Decoherence, the measurement problem, and …theory.caltech.edu/.../Decoherence....schlosshauer.pdfquantum dynamics is necessary. In his monumental book on the foundations of quantum

Decoherence, the measurement problem, and interpretations ofquantum mechanics

Maximilian Schlosshauer*

Department of Physics, University of Washington, Seattle, Washington 98195, USA

~Published 23 February 2005!

Environment-induced decoherence and superselection have been a subject of intensive research overthe past two decades, yet their implications for the foundational problems of quantum mechanics,most notably the quantum measurement problem, have remained a matter of great controversy. Thispaper is intended to clarify key features of the decoherence program, including its more recent results,and to investigate their application and consequences in the context of the main interpretiveapproaches of quantum mechanics.

CONTENTS

I. Introduction 1268

II. The Measurement Problem 1269

A. Quantum measurement scheme 1269

B. The problem of definite outcomes 1270

1. Superpositions and ensembles 1270

2. Superpositions and outcome attribution 1270

3. Objective vs subjective definiteness 1271

C. The preferred-basis problem 1272

D. The quantum-to-classical transition and decoherence 1272

III. The Decoherence Program 1273

A. Resolution into subsystems 1274

B. The concept of reduced density matrices 1274

C. A modified von Neumann measurement scheme 1275

D. Decoherence and local suppression of interference 1276

1. General formalism 1276

2. An exactly solvable two-state model for

decoherence 1276

E. Environment-induced superselection 1278

1. Stability criterion and pointer basis 1278

2. Selection of quasiclassical properties 1279

3. Implications for the preferred-basis problem 1280

4. Pointer basis vs instantaneous Schmidt

states 1281

F. Envariance, quantum probabilities, and the Born

rule 1282

1. Environment-assisted invariance 1282

2. Deducing the Born rule 1283

3. Summary and outlook 1284

IV. The Role of Decoherence in Interpretations of

Quantum Mechanics 1284

A. General implications of decoherence for

interpretations 1284

B. The standard and the Copenhagen interpretations 1285

1. The problem of definite outcomes 1285

2. Observables, measurements, and

environment-induced superselection 12863. The concept of classicality in the

Copenhagen interpretation 1287C. Relative-state interpretations 1288

1. Everett branches and the preferred-basisproblem 1288

2. Probabilities in Everett interpretations 12903. The “existential interpretation” 1290

D. Modal interpretations 12911. Property assignment based on

environment-induced superselection 12912. Property assignment based on instantaneous

Schmidt decompositions 12923. Property assignment based on

decompositions of the decohered densitymatrix 1292

4. Concluding remarks 1293E. Physical collapse theories 1293

1. The preferred-basis problem 12942. Simultaneous presence of decoherence and

spontaneous localization 12943. The tails problem 12954. Connecting decoherence and collapse

models 12955. Summary and outlook 1296

F. Bohmian mechanics 12961. Particles as fundamental entities 12962. Bohmian trajectories and decoherence 1297

G. Consistent-histories interpretations 12981. Definition of histories 12982. Probabilities and consistency 12983. Selection of histories and classicality 12994. Consistent histories of open systems 13005. Schmidt states vs pointer basis as projectors 13006. Exact vs approximate consistency 13017. Consistency and environment-induced

superselection 13018. Summary and discussion 1301

V. Concluding Remarks 1302Acknowledgments 1303References 1303*Electronic address: [email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 76, OCTOBER 2004

0034-6861/2004/76~4!/1267~39!/$50.00 ©2005 The American Physical Society1267

I. INTRODUCTION

The implications of the decoherence program for thefoundations of quantum mechanics have been the sub-ject of an ongoing debate since the first precise formu-lation of the program in the early 1980s. The key ideapromoted by decoherence is the insight that realisticquantum systems are never isolated, but are immersedin the surrounding environment and interact continu-ously with it. The decoherence program then studies,entirely within the standard quantum formalism si.e.,without adding any new elements to the mathematicaltheory or its interpretationd, the resulting formation ofquantum correlations between the states of the systemand its environment and the often surprising effects ofthese system-environment interactions. In short, deco-herence brings about a local suppression of interferencebetween preferred states selected by the interaction withthe environment.

Bub s1997d termed decoherence part of the “new or-thodoxy” of understanding quantum mechanics—as theworking physicist’s way of motivating the postulates ofquantum mechanics from physical principles. Propo-nents of decoherence called it an “historical accident”sJoos, 2000, p. 13d that the implications for quantum me-chanics and for the associated foundational problemswere overlooked for so long. Zurek s2003a, p. 717d sug-gests

The idea that the “openness” of quantum systemsmight have anything to do with the transition fromquantum to classical was ignored for a very longtime, probably because in classical physics prob-lems of fundamental importance were alwayssettled in isolated systems.

When the concept of decoherence was first introducedto the broader scientific community by Zurek’s s1991darticle in Physics Today, it elicited a series of conten-tious comments from the readership ssee the April 1993issue of Physics Todayd. In response to his critics, Zureks2003a, p. 718d states

In a field where controversy has reigned for so longthis resistance to a new paradigm fnamely, to de-coherenceg is no surprise.

Omnes s2002, p. 2d had this assessment:

The discovery of decoherence has already muchimproved our understanding of quantum mechan-ics. s…d fBgut its foundation, the range of its valid-ity and its full meaning are still rather obscure.This is due most probably to the fact that it dealswith deep aspects of physics, not yet fully investi-gated.

In particular, the question whether decoherence pro-vides, or at least suggests, a solution to the measurementproblem of quantum mechanics has been discussed forseveral years. For example, Anderson s2001, p. 492dwrites in an essay review

The last chapter s…d deals with the quantum mea-surement problem s…d. My main test, allowing meto bypass the extensive discussion, was a quick, un-successful search in the index for the word “deco-herence” which describes the process that used tobe called “collapse of the wave function.”

Zurek speaks in various places of the “apparent” or “ef-fective” collapse of the wave function induced by theinteraction with environment swhen embedded into aminimal additional interpretive frameworkd and con-cludes sZurek, 1998, p. 1793d

A “collapse” in the traditional sense is no longernecessary. s…d fTheg emergence of “objective exis-tence” ffrom decoherenceg s…d significantly re-duces and perhaps even eliminates the role of the“collapse” of the state vector.

d’Espagnat, who considers the explanation of our expe-riences si.e., of “appearances”d as the only “sure” re-quirement of a physical theory, states sd’Espagnat, 2000,p. 136d

For macroscopic systems, the appearances arethose of a classical world sno interferences etc.d,even in circumstances, such as those occurring inquantum measurements, where quantum effectstake place and quantum probabilities intervenes…d. Decoherence explains the just mentioned ap-pearances and this is a most important result. s…dAs long as we remain within the realm of merepredictions concerning what we shall observe si.e.,what will appear to usd—and refrain from statinganything concerning “things as they must be beforewe observe them”—no break in the linearity ofquantum dynamics is necessary.

In his monumental book on the foundations of quantummechanics sQMd, Auletta s2000, p. 791d concludes that

the Measurement theory could be part of the inter-pretation of QM only to the extent that it wouldstill be an open problem, and we think that this islargely no longer the case.

This is mainly so because, according to Auletta s2000, p.289d,

decoherence is able to solve practically all theproblems of Measurement which have been dis-cussed in the previous chapters.

On the other hand, even leading adherents of decoher-ence have expressed caution or even doubt that deco-herence has solved the measurement problem. Jooss2000, p. 14d writes

Does decoherence solve the measurement prob-lem? Clearly not. What decoherence tells us, isthat certain objects appear classical when they areobserved. But what is an observation? At somestage, we still have to apply the usual probabilityrules of quantum theory.

1268 Maximilian Schlosshauer: Decoherence, the measurement problem, and interpretations of quantum mechanics

Rev. Mod. Phys., Vol. 76, No. 4, October 2004

Along these lines, Kiefer and Joos s1999, p. 5d warn that

One often finds explicit or implicit statements tothe effect that the above processes are equivalentto the collapse of the wave function sor even solvethe measurement problemd. Such statements arecertainly unfounded.

In a response to Anderson’s s2001, p. 492d comment,Adler s2003, p. 136d states

I do not believe that either detailed theoretical cal-culations or recent experimental results show thatdecoherence has resolved the difficulties associatedwith quantum measurement theory.

Similarly, Bacciagaluppi s2003b, p. 3d writes

Claims that simultaneously the measurement prob-lem is real fandg decoherence solves it are con-fused at best.

Zeh asserts sJoos et al., 2003, Chap. 2d

Decoherence by itself does not yet solve the mea-surement problem s…d. This argument is nonethe-less found wide-spread in the literature. s…d Itdoes seem that the measurement problem can onlybe resolved if the Schrödinger dynamics s…d issupplemented by a nonunitary collapse s…d.

The key achievements of the decoherence program,apart from their implications for conceptual problems,do not seem to be universally understood either. Zureks1998, p. 1800d remarks

fTheg eventual diagonality of the density matrixs…d is a byproduct s…d but not the essence of de-coherence. I emphasize this because diagonality offthe density matrixg in some basis has been occa-sionally smis-dinterpreted as a key accomplishmentof decoherence. This is misleading. Any densitymatrix is diagonal in some basis. This has littlebearing on the interpretation.

These remarks show that a balanced discussion of thekey features of decoherence and their implications forthe foundations of quantum mechanics is overdue. Thedecoherence program has made great progress over thepast decade, and it would be inappropriate to ignore itsrelevance in tackling conceptual problems. However, itis equally important to realize the limitations of deco-herence in providing consistent and noncircular answersto foundational questions.

An excellent review of the decoherence program hasrecently been given by Zurek s2003ad. It deals primarilywith the technicalities of decoherence, although it con-tains some discussion on how decoherence can be em-ployed in the context of a relative-state interpretation tomotivate basic postulates of quantum mechanics. Ahelpful first orientation and overview, the entry by Bac-ciagaluppi s2003ad in the Stanford Encyclopedia of Phi-losophy, features a relatively short sin comparison to thepresent paperd introduction to the role of decoherencein the foundations of quantum mechanics, including

comments on the relationship between decoherence andseveral popular interpretations of quantum theory. Inspite of these valuable recent contributions to the litera-ture, a detailed and self-contained discussion of the roleof decoherence in the foundations of quantum mechan-ics seems still to be lacking. This review article is in-tended to fill the gap.

To set the stage, we shall first, in Sec. II, review themeasurement problem, which illustrates the key difficul-ties that are associated with describing quantum mea-surement within the quantum formalism and that are allin some form addressed by the decoherence program. InSec. III, we then introduce and discuss the main featuresof the theory of decoherence, with a particular emphasison their foundational implications. Finally, in Sec. IV, weinvestigate the role of decoherence in various interpre-tive approaches of quantum mechanics, in particularwith respect to the ability to motivate and support sordisproved possible solutions to the measurement prob-lem.

II. THE MEASUREMENT PROBLEM

One of the most revolutionary elements introducedinto physical theory by quantum mechanics is the super-position principle, mathematically founded in the linear-ity of the Hilbert-state space. If u1l and u2l are two states,then quantum mechanics tells us that any linear combi-nation au1l+bu2l also corresponds to a possible state.Whereas such superpositions of states have been experi-mentally extensively verified for microscopic systemssfor instance, through the observation of interference ef-fectsd, the application of the formalism to macroscopicsystems appears to lead immediately to severe clasheswith our experience of the everyday world. A book hasnever been observed to be in a state of being both“here” and “there” si.e., to be in a superposition of mac-roscopically distinguishable positionsd, nor does aSchrödinger cat that is a superposition of being alive anddead bear much resemblence to reality as we perceive it.The problem is, then, how to reconcile the vastness ofthe Hilbert space of possible states with the observationof a comparatively few “classical” macrosopic states, de-fined by having a small number of determinate and ro-bust properties such as position and momentum. Whydoes the world appear classical to us, in spite of its sup-posed underlying quantum nature, which would, in prin-ciple, allow for arbitrary superpositions?

A. Quantum measurement scheme

This question is usually illustrated in the context ofquantum measurement where microscopic superposi-tions are, via quantum entanglement, amplified into themacroscopic realm and thus lead to very “nonclassical”states that do not seem to correspond to what is actuallyperceived at the end of the measurement. In the idealmeasurement scheme devised by von Neumann s1932d, astypically microscopicd system S, represented by basisvectors husnlj in a Hilbert space HS, interacts with a mea-

1269Maximilian Schlosshauer: Decoherence, the measurement problem, and interpretations of quantum mechanics


surement apparatus A, described by basis vectors huanljspanning a Hilbert space HA, where the uanl are assumedto correspond to macroscopically distinguishable“pointer” positions that correspond to the outcome of ameasurement if S is in the state usnl.1

Now, if S is in a smicroscopically “unproblematic”dsuperposition oncnusnl, and A is in the initial “ready”state uarl, the linearity of the Schrödinger equation en-tails that the total system SA, assumed to be repre-sented by the Hilbert product space HS ^ HA, evolvesaccording to

Son

cnusnlDuarl→t

on

cnusnluanl . s2.1d

This dynamical evolution is often referred to as a pre-measurement in order to emphasize that the process de-scribed by Eq. s2.1d does not suffice to directly concludethat a measurement has actually been completed. This isso for two reasons. First, the right-hand side is a super-position of system-apparatus states. Thus, without sup-plying an additional physical process ssay, some collapsemechanismd or giving a suitable interpretation of such asuperposition, it is not clear how to account, given thefinal composite state, for the definite pointer positionsthat are perceived as the result of an actualmeasurement—i.e., why do we seem to perceive thepointer to be in one position uanl but not in a superpo-sition of positions? This is the problem of definite out-comes. Second, the expansion of the final compositestate is in general not unique, and therefore the mea-sured observable is not uniquely defined either. This isthe problem of the preferred basis. In the literature, thefirst difficulty is typically referred to as the measurementproblem, but the preferred-basis problem is at leastequally important, since it does not make sense even toinquire about specific outcomes if the set of possible out-comes is not clearly defined. We shall therefore regardthe measurement problem as composed of both theproblem of definite outcomes and the problem of thepreferred basis, and discuss these components in moredetail in the following.

B. The problem of definite outcomes

1. Superpositions and ensembles

The right-hand side of Eq. s2.1d implies that after thepremeasurement the combined system SA is left in apure state that represents a linear superposition ofsystem-pointer states. It is a well-known and importantproperty of quantum mechanics that a superposition ofstates is fundamentally different from a classical en-

semble of states, where the system actually is in only oneof the states but we simply do not know in which sthis isoften referred to as an “ignorance-interpretable,” or“proper” ensembled.

This can be shown explicitly, especially on microscopicscales, by performing experiments that lead to the directobservation of interference patterns instead of the real-ization of one of the terms in the superposed pure state,for example, in a setup where electrons pass individuallysone at a timed through a double slit. As is well known,this experiment clearly shows that, within the standardquantum-mechanical formalism, the electron must notbe described by either one of the wave functions de-scribing the passage through a particular slit sc1 or c2d,but only by the superposition of these wave functionssc1+c2d. This is so because the correct density distribu-tion % of the pattern on the screen is not given by thesum of the squared wave functions describing the addi-tion of individual passages through a single slit s%= uc1u2+ uc2u2d, but only by the square of the sum of theindividual wave functions s%= uc1+c2u2d.

Put differently, if an ensemble interpretation could beattached to a superposition, the latter would simply rep-resent an ensemble of more fundamentally determinedstates, and based on the additional knowledge broughtabout by the results of measurements, we could simplychoose a subensemble consisting of the definite pointerstate obtained in the measurement. But then, since thetime evolution has been strictly deterministic accordingto the Schrödinger equation, we could backtrack thissubensemble in time and thus also specify the initialstate more completely s“postselection”d, and thereforethis state necessarily could not be physically identical tothe initially prepared state on the left-hand side of Eq.s2.1d.

2. Superpositions and outcome attribution

In the standard s“orthodox”d interpretation of quan-tum mechanics, an observable corresponding to a physi-cal quantity has a definite value if and only if the systemis in an eigenstate of the observable; if the system is,however, in a superposition of such eigenstates, as in Eq.s2.1d, it is, according to the orthodox interpretation,meaningless to speak of the state of the system as havingany definite value of the observable at all. sThis is fre-quently referred to as the so-called eigenvalue-eigenstatelink, or “e-e link” for short.d The e-e link, however, is byno means forced upon us by the structure of quantummechanics or by empirical constraints sBub, 1997d. Theconcept of sclassicald “values” that can be ascribedthrough the e-e link based on observables and the exis-tence of exact eigenstates of these observables hastherefore frequently been either weakened or altogetherabandonded. For instance, outcomes of measurementsare typically registered in position space spointer posi-tions, etc.d, but there exist no exact eigenstates of theposition operator, and the pointer states are never ex-actly mutually orthogonal. One might then sexplicitly orimplicitlyd promote a “fuzzy” e-e link, or give up the

1Note that von Neumann’s scheme is in sharp contrast to theCopenhagen interpretation, where measurement is not treatedas a system-apparatus interaction described by the usualquantum-mechanical formalism, but instead as an independentcomponent of the theory, to be represented entirely in funda-mentally classical terms.



concept of observables and values entirely and directlyinterpret the time-evolved wave functions sworking inthe Schrödinger pictured and the corresponding densitymatrices. Also, if it is regarded as sufficient to explainour perceptions rather than describe the “absolute”state of the entire universe ssee the argument belowd,one might only require that the sexact or fuzzyd e-e linkhold in a “relative” sense, i.e., for the state of the rest ofthe universe relative to the state of the observer.

Then, to solve the problem of definite outcomes, someinterpretations sfor example, modal interpretations andrelative-state interpretationsd interpret the final compos-ite state in such a way as to explain the existence, or atleast the subjective perception, of “outcomes” even ifthis state has the form of a superposition. Other inter-pretations attempt to solve the measurement problemby modifying the strictly unitary Schrödinger dynamics.Most prominently, the orthodox interpretation postu-lates a collapse mechanism that transforms a pure-statedensity matrix into an ignorance-interpretable ensembleof individual states sa “proper mixture”d. Wave-functioncollapse theories add stochastic terms to the Schrödingerequation that induce an effective salbeit only approxi-mated collapse for states of macroscopic systems sPearle,1979, 1999; Gisin, 1984; Ghirardi et al., 1986d, whileother authors have suggested that collapse occurs at thelevel of the mind of a conscious observer sWigner, 1963;Stapp, 1993d. Bohmian mechanics, on the other hand,upholds a unitary time evolution of the wave function,but introduces an additional dynamical law that explic-itly governs the always-determinate positions of all par-ticles in the system.

3. Objective vs subjective definiteness

In general, smacroscopicd definiteness—and thus a so-lution to the problem of outcomes in the theory of quan-tum measurement—can be achieved either on an onto-logical sobjectived or an observational ssubjectived level.Objective definiteness aims at ensuring “actual” defi-niteness in the macroscopic realm, whereas subjectivedefiniteness only attempts to explain why the macro-scopic world appears to be definite—and thus does notmake any claims about definiteness of the underlyingphysical reality swhatever this reality might bed. Thisraises the question of the significance of this distinctionwith respect to the formation of a satisfactory theory ofthe physical world. It might appear that a solution to themeasurement problem based on ensuring subjective, butnot objective, definiteness is merely good “for all prac-tical purposes”—abbreviated, rather disparagingly, as“FAPP” by Bell s1990d—and thus not capable of solvingthe “fundamental” problem that would seem relevant tothe construction of the “precise theory” that Bell de-manded so vehemently.

It seems to the author, however, that this criticism isnot justified, and that subjective definiteness should beviewed on a par with objective definiteness with respectto a satisfactory solution to the measurement problem.

We demand objective definiteness because we experi-ence definiteness on the subjective level of observation,and it should not be viewed as an a priori requirementfor a physical theory. If we knew independently of ourexperience that definiteness existed in nature, subjectivedefiniteness would presumably follow as soon as we hademployed a simple model that connected the “external”physical phenomena with our “internal” perceptual andcognitive apparatus, where the expected simplicity ofsuch a model can be justified by referring to the pre-sumed identity of the physical laws governing externaland internal processes. But since knowledge is based onexperience, that is, on observation, the existence of ob-jective definiteness could only be derived from the ob-servation of definiteness. And, moreover, observationtells us that definiteness is in fact not a universal prop-erty of nature, but rather a property of macroscopic ob-jects, where the borderline to the macroscopic realm isdifficult to draw precisely; mesoscopic interference ex-periments have demonstrated clearly the blurriness ofthe boundary. Given the lack of a precise definition ofthe boundary, any demand for fundamental definitenesson the objective level should be based on a much deeperand more general commitment to a definiteness that ap-plies to every physical entity sor systemd across theboard, regardless of spatial size, physical property, andthe like.

Therefore, if we realize that the often deeply felt com-mitment to a general objective definiteness is only basedon our experience of macroscopic systems, and that thisdefiniteness in fact fails in an observable manner for mi-croscopic and even certain mesoscopic systems, the au-thor sees no compelling grounds on which objectivedefiniteness must be demanded as part of a satisfactoryphysical theory, provided that the theory can account forsubjective, observational definiteness in agreement withour experience. Thus the author suggests that the samelegitimacy be attributed to proposals for a solution ofthe measurement problem that achieve “only” subjec-tive but not objective definiteness—after all, the mea-surement problem arises solely from a clash of our ex-perience with certain implications of the quantumformalism. d’Espagnat s2000, pp. 134 and 135d has advo-cated a similar viewpoint:

The fact that we perceive such “things” as macro-scopic objects lying at distinct places is due, partlyat least, to the structure of our sensory and intel-lectual equipment. We should not, therefore, takeit as being part of the body of sure knowledge thatwe have to take into account for defining a quan-tum state. s…d In fact, scientists most rightly claimthat the purpose of science is to describe humanexperience, not to describe “what really is”; and aslong as we only want to describe human experi-ence, that is, as long as we are content with beingable to predict what will be observed in all possiblecircumstances s…d we need not postulate the



existence—in some absolute sense—of unobservedsi.e., not yet observedd objects lying at definiteplaces in ordinary 3-dimensional space.

C. The preferred-basis problem

The second difficulty associated with quantum mea-surement is known as the preferred-basis problem,which demonstrates that the measured observable is ingeneral not uniquely defined by Eq. s2.1d. For any choiceof system states husnlj, we can find corresponding appa-ratus states huanlj, and vice versa, to equivalently rewritethe final state emerging from the premeasurement inter-action, i.e., the right-hand side of Eq. s2.1d. In general,however, for some choice of apparatus states the corre-sponding new system states will not be mutually or-thogonal, so that the observable associated with thesestates will not be Hermitian, which is usually not desiredshowever, not forbidden—see the discussion by Zurek,2003ad. Conversely, to ensure distinguishable outcomes,we must, in general, require the sat least approximatedorthogonality of the apparatus spointerd states, and itthen follows from the biorthogonal decomposition theo-rem that the expansion of the final premeasurementsystem-apparatus state of Eq. s2.1d,

ucl = on

cnusnluanl , s2.2d

is unique, but only if all coefficients cn are distinct. Oth-erwise, we can in general rewrite the state in terms ofdifferent state vectors,

ucl = on

cn8usn8luan8l , s2.3d

such that the same postmeasurement state seems to cor-respond to two different measurements, that is, of the

observables A=onlnusnlksnu and B=onln8usn8lksn8u of the

system, respectively, although in general A and B do notcommute.

As an example, consider a Hilbert space H=H1 ^ H2where H1 and H2 are two-dimensional spin spaces withstates corresponding to spin up or spin down along agiven axis. Suppose we are given an entangled spin stateof the Einstein-Podolsky-Rosen form sEinstein et al.,1935d

ucl =1Î2

suz + l1uz − l2 − uz − l1uz + l2d , s2.4d

where uz± l1,2 represents the eigenstates of the observ-able sz corresponding to spin up or spin down along thez axis of the two systems 1 and 2. The state ucl can,however, equivalently be expressed in the spin basis cor-responding to any other orientation in space. For ex-ample, when using the eigenstates ux± l1,2 of the observ-able sx swhich represents a measurement of the spinorientation along the x axisd as basis vectors, we get

ucl =1Î2

sux + l1ux − l2 − ux − l1ux + l2d . s2.5d

Now suppose that system 2 acts as a measuring devicefor the spin of system 1. Then Eqs. s2.4d and s2.5d implythat the measuring device has established a correlationwith both the z and the x spin of system 1. This meansthat, if we interpret the formation of such a correlationas a measurement in the spirit of the von Neumannscheme swithout assuming a collapsed, our apparatusssystem 2d could be considered as having measured alsothe x spin once it has measured the z spin, and viceversa—in spite of the noncommutativity of the corre-sponding spin observables sz and sx. Moreover, since wecan rewrite Eq. s2.4d in infinitely many ways, it appearsthat once the apparatus has measured the spin of system1 along one direction, it can also be regarded as havingmeasured the spin along any other direction, again inapparent contradiction with quantum mechanics due tothe noncommutativity of the spin observables corre-sponding to different spatial orientations.

It thus seems that quantum mechanics has nothing tosay about which observablessd of the system is sared be-ing recorded, via the formation of quantum correlations,by the apparatus. This can be stated in a general theo-rem sZurek, 1981; Auletta, 2000d: When quantum me-chanics is applied to an isolated composite object con-sisting of a system S and an apparatus A, it cannotdetermine which observable of the system has beenmeasured—in obvious contrast to our experience of theworkings of measuring devices that seem to be “de-signed” to measure certain quantities.

D. The quantum-to-classical transition and decoherence

In essence, as we have seen above, the measurementproblem deals with the transition from a quantum world,described by essentially arbitrary linear superpositionsof state vectors, to our perception of “classical” states inthe macroscopic world, that is, a comparatively smallsubset of the states allowed by the quantum-mechanicalsuperposition principle, having only a few, but determi-nate and robust, properties, such as position, momen-tum, etc. The question of why and how our experienceof a “classical” world emerges from quantum mechanicsthus lies at the heart of the foundational problems ofquantum theory.

Decoherence has been claimed to provide an explana-tion for this quantum-to-classical transition by appealingto the ubiquitous immersion of virtually all physical sys-tems in their environment s“environmental monitor-ing”d. This trend can also be read off nicely from thetitles of some papers and books on decoherence, for ex-ample, “The emergence of classical properties throughinteraction with the environment” sJoos and Zeh, 1985d,“Decoherence and the transition from quantum to clas-sical” sZurek, 1991d, and “Decoherence and the appear-ance of a classical world in quantum theory” sJoos et al.,2003d. We shall critically investigate in this paper to what



extent the appeal to decoherence for an explanation ofthe quantum-to-classical transition is justified.

III. THE DECOHERENCE PROGRAM

As remarked earlier, the theory of decoherence isbased on a study of the effects brought about by theinteraction of physical systems with their environment.In classical physics, the environment is usually viewed asa kind of disturbance, or noise, that perturbs the systemunder consideration in such a way as to negatively influ-ence the study of its “objective” properties. Thereforescience has established the idealization of isolated sys-tems, with experimental physics aiming at eliminatingany outer sources of disturbance as much as possible inorder to discover the “true” underlying nature of thesystem under study.

The distinctly nonclassical phenomenon of quantumentanglement, however, has demonstrated that the cor-relations between two systems can be of fundamentalimportance and can lead to properties that are notpresent in the individual systems.2 The earlier view ofphenomena arising from quantum entanglement as“paradoxa” has generally been replaced by the recogni-tion of entanglement as a fundamental property of na-ture.

The decoherence program3 is based on the idea thatsuch quantum correlations are ubiquitous; that nearlyevery physical system must interact in some way with itsenvironment sfor example, with the surrounding pho-tons that then create the visual experience within theobserverd, which typically consists of a large number ofdegrees of freedom that are hardly ever fully controlled.Only in very special cases of typically microscopicsatomicd phenomena, so goes the claim of the decoher-ence program, is the idealization of isolated systems ap-plicable so that the predictions of linear quantum me-chanics si.e., a large class of superpositions of statesd canactually be observationally confirmed. In the majority ofthe cases accessible to our experience, however, interac-tion with the environment is so dominant as to precludethe observation of the “pure” quantum world, imposingeffective superselection rules sWick et al., 1952, 1970;Galindo et al., 1962; Wightman, 1995; Cisnerosy et al.,1998; Giulini, 2000d onto the space of observable statesthat lead to states corresponding to the “classical” prop-erties of our experience. Interference between suchstates gets locally suppressed and is thus claimed to be-come inaccessible to the observer.

Probably the most surprising aspect of decoherence isthe effectiveness of the system-environment interac-tions. Decoherence typically takes place on extremelyshort time scales and requires the presence of only a

minimal environment sJoos and Zeh, 1985d. Due to thelarge number of degrees of freedom of the environment,it is usually very difficult to undo system-environmententanglement, which has been claimed as a source of ourimpression of irreversibility in nature ssee, for example,Zurek, 1982, 2003a; Zurek and Paz, 1994; Kiefer andJoos, 1999; Zeh, 2001d. In general, the effect of decoher-ence increases with the size of the system sfrom micro-scopic to macroscopic scalesd, but it is important to notethat there exist, admittedly somewhat exotic, examplesfor which the decohering influence of the environmentcan be sufficiently shielded to lead to mesoscopic andeven macroscopic superpositions. One such examplewould be the case of superconducting quantum interfer-ence devices sSQUID’sd, in which superpositions of mac-roscopic currents become observable. Conversely, somemicroscopic systems sfor instance, certain chiral mol-ecules that exist in different distinct spatial configura-tionsd can be subject to remarkably strong decoherence.

The decoherence program has dealt with the follow-ing two main consequences of environmental interac-tion:

s1d Environment-induced decoherence: The fast localsuppression of interference between different statesof the system. However, since only unitary time evo-lution is employed, global phase coherence is notactually destroyed—it becomes absent from the lo-cal density matrix that describes the system alone,but remains fully present in the total system-environment composition.4 We shall discussenvironment-induced local decoherence in more de-tail in Sec. III.D.

s2d Environment-induced superselection: The selectionof preferred sets of states, often referred to as“pointer states,” that are robust sin the sense of re-taining correlations over timed in spite of their im-mersion in the environment. These states are deter-mined by the form of the interaction between thesystem and its environment and are suggested tocorrespond to the “classical” states of our experi-ence. We shall consider this mechanism in Sec. III.E.

Another, more recent aspect of the decoherence pro-gram, termed enviroment-assisted invariance or “envari-ance,” was introduced by Zurek s2003a, 2003b, 2004bdand further developed in Zurek s2004ad. In particular,Zurek used envariance to explain the emergence ofprobabilities in quantum mechanics and to derive Born’srule based on certain assumptions. We shall review en-variance and Zurek’s derivation of the Born rule in Sec.III.F.

Finally, let us emphasize that decoherence arises froma direct application of the quantum-mechanical formal-ism to a description of the interaction of a physical sys-

2Broadly speaking, this means that the squantum-mechanicaldwhole is different from the sum of its parts.

3For key ideas and concepts, see Zeh s1970, 1973, 1995, 1997,2000d; Zurek s1981, 1982, 1991, 1993, 2003ad; Kübler and Zehs1973d; Joos and Zeh s1985d; Joos et al. s2003d.

4Note that the persistence of coherence in the total state isimportant to ensure the possibility of describing special casesin which mesoscopic or macrosopic superpositions have beenexperimentally realized.



tem with its environment. By itself, decoherence istherefore neither an interpretation nor a modification ofquantum mechanics. Yet the implications of decoher-ence need to be interpreted in the context of the differ-ent interpretations of quantum mechanics. Also, sincedecoherence effects have been studied extensively inboth theoretical models and experiments sfor a survey,see, for example, Joos et al., 2003; Zurek, 2003ad, theirexistence can be taken as a well-confirmed fact.

A. Resolution into subsystems

Note that decoherence derives from the presupposi-tion of the existence and the possibility of a division ofthe world into “systemssd” and “environment.” In thedecoherence program, the term “environment” is usu-ally understood as the “remainder” of the system, in thesense that its degrees of freedom are typically not scan-not be, do not need to bed controlled and are not di-rectly relevant to the observation under considerationsfor example, the many microsopic degrees of freedomof the systemd, but that nonetheless the environment in-cludes “all those degrees of freedom which contributesignificantly to the evolution of the state” of the systemsZurek, 1981, p. 1520d.

This system-environment dualism is generally associ-ated with quantum entanglement, which always de-scribes a correlation between parts of the universe. Aslong as the universe is not resolved into individual sub-systems, there is no measurement problem: the statevector uCl of the entire universe5 evolves deterministi-cally according to the Schrödinger equation i"s] /]tduCl=HuCl, which poses no interpretive difficulty. Onlywhen we decompose the total Hilbert-state space H ofthe universe into a product of two spaces H1 ^ H2, andaccordingly form the joint-state vector uCl= uC1luC2l,and want to ascribe an individual state sbesides the jointstate that describes a correlationd to one of the two sys-tems ssay, the apparatusd, does the measurement prob-lem arise. Zurek s2003a, p. 718d puts it like this:

In the absence of systems, the problem of interpre-tation seems to disappear. There is simply no needfor “collapse” in a universe with no systems. Ourexperience of the classical reality does not apply tothe universe as a whole, seen from the outside, butto the systems within it.

Moreover, terms like “observation,” “correlation,” and“interaction” will naturally make little sense without adivision into systems. Zeh has suggested that the localityof the observer defines an observation in the sense thatany observation arises from the ignorance of a part ofthe universe; and that this also defines the “facts” thatcan occur in a quantum system. Landsman s1995, pp. 45and 46d argues similarly:

The essence of a “measurement,” “fact” or“event” in quantum mechanics lies in the non-observation, or irrelevance, of a certain part of thesystem in question. s…d A world without parts de-clared or forced to be irrelevant is a world withoutfacts.

However, the assumption of a decomposition of the uni-verse into subsystems—as necessary as it appears to befor the emergence of the measurement problem and forthe definition of the decoherence program—is definitelynontrivial. By definition, the universe as a whole is aclosed system, and therefore there are no “unobserveddegrees of freedom” of an external environment whichwould allow for the application of the theory of decoher-ence to determine the space of quasiclassical observ-ables of the universe in its entirety. Also, there exists nogeneral criterion for how the total Hilbert space is to bedivided into subsystems, while at the same time much ofwhat is called a property of the system will depend on itscorrelation with other systems. This problem becomesparticularly acute if one would like decoherence notonly to motivate explanations for the subjective percep-tion of classicality sas in Zurek’s “existential interpreta-tion”; see Zurek, 1993, 1998, 2003a, and Sec. IV.C be-lowd, but moreover to allow for the definition ofquasiclassical “macrofacts.” Zurek s1998, p. 1820d admitsthis severe conceptual difficulty:

In particular, one issue which has been often takenfor granted is looming big, as a foundation of thewhole decoherence program. It is the question ofwhat are the “systems” which play such a crucialrole in all the discussions of the emergent classical-ity. s…d fAg compelling explanation of what are thesystems—how to define them given, say, the over-all Hamiltonian in some suitably large Hilbertspace—would be undoubtedly most useful.

A frequently proposed idea is to abandon the notion ofan “absolute” resolution and instead postulate the in-trinsic relativity of the distinct state spaces and proper-ties that emerge through the correlation between theserelatively defined spaces ssee, for example, the propos-als, unrelated to decoherence, of Everett, 1957, Mermin,1998a, 1998b; and Rovelli, 1996d. This relative view ofsystems and correlations has counterintuitive, in thesense of nonclassical, implications. However, as in thecase of quantum entanglement, these implications neednot be taken as paradoxa that demand further resolu-tion. Accepting some properties of nature as counterin-tuitive is indeed a satisfactory path to take in order toarrive at a description of nature that is as complete andobjective as is allowed by the range of our experienceswhich is based on inherently local observationsd.

B. The concept of reduced density matrices

Since reduced density matrices are a key tool of deco-herence, it will be worthwile to briefly review their basicproperties and interpretation in the following. The con-

5If we dare to postulate this total state—see counterargu-ments by Auletta s2000d.



cept of reduced density matrices emerged in the earliestdays of quantum mechanics sLandau, 1927; von Neu-mann, 1932; Furry, 1936; for some historical remarks, seePessoa, 1998d. In the context of a system of two en-tangled systems in a pure state of the Einstein-Podolsky-Rosen-type,

ucl =1Î2

su + l1u− l2 − u− l1u + l2d , s3.1d

it had been realized early that for an observable O that

pertains only to system 1, O=O1 ^ I2, the pure-state den-sity matrix r= uclkcu yields, according to the trace rule

kOl=TrsrOd and given the usual Born rule for calculat-ing probabilities, exactly the same statistics as the re-duced density matrix r1 obtained by tracing over thedegrees of freedom of system 2 si.e., the states u+ l2 andu−l2d,

r1 = Tr2uclkcu = 2k+ uclkcu + l2 + 2k− uclkcu− l2, s3.2d

since it is easy to show that, for this observable O,

kOlc = TrsrOd = Tr1sr1O1d . s3.3d

This result holds in general for any pure state ucl=oiaiufil1ufil2¯ ufilN of a resolution of a system into Nsubsystems, where the hufiljj are assumed to form ortho-normal basis sets in their respective Hilbert spaces

Hj , j=1¯N. For any observable O that pertains only to

system j, O= I1 ^ I2 ^ ¯ ^ Ij−1 ^ Oj ^ Ij+1 ^ ¯ ^ IN, the

statistics of O generated by applying the trace rule willbe identical regardless of whether we use the pure-statedensity matrix r= uclkcu or the reduced density matrix

rj=Tr1,…,j−1,j+1,…,Nuclkcu, since again kOl=TrsrOd=TrjsrjOjd.

The typical situation in which the reduced density ma-trix arises is this: Before a premeasurement-type inter-action, the observer knows that each individual system isin some sunknownd pure state. After the interaction, i.e.,after the correlation between the systems is established,the observer has access to only one of the systems, say,system 1; everything that can be known about the stateof the composite system must therefore be derived frommeasurements on system 1, which will yield the possibleoutcomes of system 1 and their probability distribution.All information that can be extracted by the observer isthen, exhaustively and correctly, contained in the re-duced density matrix of system 1, assuming that theBorn rule for quantum probabilities holds.

Let us return to the Einstein-Podolsky-Rosen-type ex-ample, Eqs. s3.1d and s3.2d. If we assume that the statesof system 2 are orthogonal, 2k+u−l2=0, r1 becomes diag-onal,

r1 = Tr2uclkcu =12

su + k+ ud1 +12

su− lk− ud1. s3.4d

But this density matrix is formally identical to the den-sity matrix that would be obtained if system 1 were in a

mixed state, i.e., in either one of the two states u+ l1 andu−l1 with equal probabilties—as opposed to the superpo-sition ucl, in which both terms are considered present,which could in principle be confirmed by suitable inter-ference experiments. This implies that a measurement ofan observable that only pertains to system 1 cannot dis-criminate between the two cases, pure vs mixed state.6

However, note that the formal identification of thereduced density matrix with a mixed-state density matrixis easily misinterpreted as implying that the state of thesystem can be viewed as mixed too ssee also the discus-sion by d’Espagnat, 1988d. Density matrices are only acalculational tool for computing the probability distribu-tion of a set of possible outcomes of measurements; theydo not specify the state of the system.7 Since the twosystems are entangled and the total composite system isstill described by a superposition, it follows from thestandard rules of quantum mechanics that no individualdefinite state can be attributed to one of the systems.The reduced density matrix looks like a mixed-statedensity matrix because, if one actually measured an ob-servable of the system, one would expect to get a defi-nite outcome with a certain probability; in terms of mea-surement statistics, this is equivalent to the situation inwhich the system is in one of the states from the set ofpossible outcomes from the beginning, that is, before themeasurement. As Pessoa s1998, p. 432d puts it, “taking apartial trace amounts to the statistical version of the pro-jection postulate.”

C. A modified von Neumann measurement scheme

Let us reconsider the von Neumann model for idealquantum-mechanical measurement, Eq. s3.5d, but nowwith the environment included. We shall denote the en-vironment by E and represent its state before the mea-surement interaction by the initial state vector ue0l in aHilbert space HE. As usual, let us assume that the statespace of the composite object system-apparatus-environment is given by the tensor product of the indi-vidual Hilbert spaces, HS ^ HA ^ HE. The linearity of theSchrödinger equation then yields the following time evo-lution of the entire system SAE,

Son

cnusnlDuarlue0l→s1d So

ncnusnluanlDue0l

→s2d

on

cnusnluanluenl , s3.5d

where the uenl are the states of the environment associ-

6As discussed by Bub s1997, pp. 208–210d, this result alsoholds for any observable of the composite system that factor-izes into the form O=O1 ^ O2, where O1 and O2 do not com-mute with the projection operators su± lk±ud1 and su± lk±ud2, re-spectively.

7In this context we note that any nonpure density matrix canbe written in many different ways, demonstrating that any par-tition in a particular ensemble of quantum states is arbitrary.



ated with the different pointer states uanl of the measur-ing apparatus. Note that while for two subsystems, say, Sand A, there always exists a diagonal s“Schmidt”d de-composition of the final state of the form oncnusnluanl, forthree subsystems sfor example, S , A, and Ed, a decom-position of the form oncnusnluanluenl is not always pos-sible. This implies that the total Hamiltonian that in-duces a time evolution of the above kind, Eq. s3.5d, mustbe of a special form.8

Typically, the uenl will be product states of many mi-crosopic subsystem states u«nli corresponding to the in-dividual parts that form the environment, i.e., uenl= u«nl1u«nl2u«nl3¯. We see that a nonseparable and inmost cases, for all practical purposes, irreversible sdue tothe enormous number of degrees of freedom of the en-vironmentd correlation has been established between thestates of the system-apparatus combination SA and thedifferent states of the environment E. Note that Eq. s3.5dalso implies that the environment has recorded the stateof the system—and, equivalently, the state of the system-apparatus composition. The environment, composed ofmany subsystems, thus acts as an amplifying, higher-order measuring device.

D. Decoherence and local suppression of interference

Interaction with the environment typically leads to arapid vanishing of the diagonal terms in the local densitymatrix describing the probability distribution for theoutcomes of measurements on the system. This effecthas become known as environment-induced decoher-ence, and it has also frequently been claimed to imply atleast a partial solution to the measurement problem.

1. General formalism

In Sec. III.B, we have already introduced the conceptof local sor reducedd density matrices and pointed outsome caveats on their interpretation. In the context ofthe decoherence program, reduced density matricesarise as follows. Any observation will typically be re-stricted to the system-apparatus component, SA, whilethe many degrees of freedom of the environment E re-main unobserved. Of course, typically some degrees offreedom of the environment will always be included inour observation se.g., some of the photons scattered offthe apparatusd and we shall accordingly include them inthe “observed part SA of the universe.” The crucialpoint is that there still remains a comparatively largenumber of environmental degrees of freedom that willnot be observed directly.

Suppose then that the operator OSA represents an ob-

servable of SA only. Its expectation value kOSAl is givenby

kOSAl = TrsrSAEfOSA ^ IEgd = TrSAsrSAOSAd , s3.6d

where the density matrix rSAE of the total SAE combi-nation,

rSAE = omn

cmcn*usmluamluemlksnukanukenu , s3.7d

has, for all purposes of statistical prediction, been re-placed by the local sor reducedd density matrix rSA, ob-tained by “tracing out the unobserved degrees of theenvironment,” that is,

rSA = TrEsrSAEd = omn

cmcn*usmluamlksnukanukenuueml . s3.8d

So far, rSA contains characteristic interference termsusmluamlksnukanu, mÞn, since we cannot assume from theoutset that the basis vectors ueml of the environment arenecessarily mutually orthogonal, i.e., that ken ueml=0 ifmÞn. Many explicit physical models for the interactionof a system with the environment ssee Sec. III.D.2 belowfor a simple exampled, however, have shown that due tothe large number of subsystems that compose the envi-ronment, the pointer states uenl of the environment rap-idly approach orthogonality, ken uemlstd→dn,m, such thatthe reduced density matrix rSA becomes approximatelyorthogonal in the “pointer basis” huanlj; that is,

rSA→t

rSAd < o

nucnu2usnluanlksnukanu

= on

ucnu2PnsSd

^ PnsAd. s3.9d

Here, PnsSd and Pn

sAd are the projection operators onto theeigenstates of S and A, respectively. Therefore the inter-ference terms have vanished in this local representation,i.e., phase coherence has been locally lost. This is pre-cisely the effect referred to as environment-induced de-coherence. The decohered local density matrices de-scribing the probability distribution of the outcomes of ameasurement on the system-apparatus combination areformally sapproximatelyd identical to the correspondingmixed-state density matrix. But as we pointed out in Sec.III.B, we must be careful in interpreting this state ofaffairs, since full coherence is retained in the total den-sity matrix rSAE.

2. An exactly solvable two-state model for decoherence

To see how the approximate mutual orthogonality ofthe environmental state vectors arises, let us discuss asimple model first introduced by Zurek s1982d. Considera system S with two spin states hu⇑ l , u⇓ lj that interactswith an environment E described by a collection of Nother two-state spins represented by hu↑kl , u↓klj, k

=1¯N. The self-Hamiltonians HS and HE and the self-

8For an example of such a Hamiltonian, see the model ofZurek s1981, 1982d and its outline in Sec. III.D.2 below. For acritical comment regarding limitations on the form of the evo-lution operator and the possibility of a resulting disagreementwith experimental evidence, see Pessoa s1998d.



interaction Hamiltonian HEE of the environment aretaken to be equal to zero. Only the interaction Hamil-

tonian HSE that describes the coupling of the spin of thesystem to the spins of the environment is assumed to benonzero and of the form

HSE = su ⇑ lk⇑ u − u ⇓ lk⇓ ud ^ ok

gksu↑klk↑ku − u↓klk↓kud

^

k8ÞkIk8, s3.10d

where the gk are coupling constants and Ik= su↑klk↑ku+ u↓klk↓kud is the identity operator for the kth environ-mental spin. Applied to the initial state before the inter-action is turned on,

ucs0dl = sau ⇑ l + bu ⇓ ld ^k=1

N

saku↑kl + bku↓kld , s3.11d

this Hamiltonian yields a time evolution of the stategiven by

ucstdl = au ⇑ luE⇑stdl + bu ⇓ luE⇓stdl , s3.12d

where the two environmental states uE⇑stdl and uE⇓stdlare

uE⇑stdl = uE⇓s− tdl = ^k=1

N

sakeigktu↑kl + bke−igktu↓kld .

s3.13d

The reduced density matrix rSstd=TrEsucstdlkcstdud isthen

rSstd = uau2u ⇑ lk⇑ u + ubu2u ⇓ lk⇓ u + zstdab*u ⇑ lk⇓ u

+ z*stda*bu ⇓ lk⇑ u , s3.14d

where the interference coefficient zstd which determinesthe weight of the off-diagonal elements in the reduceddensity matrix is given by

zstd = kE⇑stduE⇓stdl = pk=1

N

suaku2eigkt + ubku2e−igktd , s3.15d

and thus

uzstdu2 = pk=1

N

h1 + fsuaku2 − ubku2d2 − 1g sin22gktj . s3.16d

At t=0, zstd=1, i.e., the interference terms are fullypresent, as expected. If uaku2=0 or 1 for each k, i.e., if theenvironment is in an eigenstate of the interaction Hamil-

tonian HSE of the type u↑1lu↑2lu↓3l¯ u↑Nl, and/or if 2gkt=mp sm=0,1,…d, then zstd2;1, so coherence is retainedover time. However, under realistic circumstances, wecan typically assume a random distribution of the initialstates of the environment si.e., of coefficients ak ,bkd andof the coupling coefficients gk. Then, in the long-timeaverage,

kuzstdu2lt→` . 2−Npk=1

N

f1 + suaku2 − ubku2d2g →N→`

0, s3.17d

so the off-diagonal terms in the reduced density matrixbecome strongly damped for large N.

It can also be shown directly that, given very generalassumptions about the distribution of the couplings gksnamely, requiring their initial distribution to have finitevarianced, zstd exhibits a Gaussian time dependence ofthe form zstd~eiAte−B2t2/2, where A and B are real con-stants sZurek et al., 2003d. For the special case in whichak=a and gk=g for all k, this behavior of zstd can beimmediately seen by first rewriting zstd as the binomialexpansion

zstd = suau2eigt + ubu2e−igtdN = ol=0

N SN

lDuau2lubu2sN−ldeigs2l−Ndt.

s3.18d

For large N, the binomial distribution can then be ap-proximated by a Gaussian,

SN

lDuau2lubu2sN−ld <

e−sl − Nuau2d2/s2Nuau2ubu2d

Î2pNuau2ubu2, s3.19d

in which case zstd becomes

zstd = ol=0

Ne−sl − Nuau2d2/s2Nuau2ubu2d

Î2pNuau2ubu2eigs2l−Ndt, s3.20d

that is, zstd is the Fourier transform of an sapproxi-matelyd Gaussian distribution and is therefore itself sap-proximatelyd Gaussian.

Detailed model calculations, in which the environ-ment is typically represented by a more sophisticatedmodel consisting of a collection of harmonic oscillatorssCaldeira and Leggett, 1983; Unruh and Zurek, 1989;Hu et al., 1992; Zurek et al., 1993; Joos et al., 2003;Zurek, 2003ad, have shown that the damping occurs onextremely short decoherence time scales tD, which aretypically many orders of magnitude shorter than thethermal relaxation. Even microscopic systems such aslarge molecules are rapidly decohered by the interactionwith thermal radiation on a time scale that is muchshorter than any practical observation could resolve; formesoscopic systems such as dust particles, the 3K cosmicmicrowave background radiation is sufficient to yieldstrong and immediate decoherence sJoos and Zeh, 1985;Zurek, 1991d.

Within tD, uzstdu approaches zero and remains close tozero, fluctuating with an average standard deviation ofthe random-walk-type s,ÎN sZurek, 1982d. However,the multiple periodicity of zstd implies that coherence,and thus the purity of the reduced density matrix, willreappear after a certain time tr, which can be shown tobe very long and of the Poincaré-type with tr,N!. Formacroscopic environments of realistic but finite sizes, trcan exceed the lifetime of the universe sZurek, 1982d,but nevertheless always remains finite.



From a conceptual point of view, recurrence of coher-ence is of little relevance. The recurrence time couldonly be infinitely long in the hypothetical case of an in-finitely large environment. In this situation, off-diagonalterms in the reduced density matrix would be irrevers-ibly damped and lost in the limit t→`, which has some-times been regarded as describing a physical collapse ofthe state vector sHepp, 1972d. But the assumption ofinfinite sizes and times is never realized in nature sBell,1975d, nor can information ever be truly lost sasachieved by a “true” state vector collapsed through uni-tary time evolution—full coherence is always retained atall times in the total density matrix rSAEstd= ucstdlkcstdu.

We can therefore state the general conclusion that,except for exceptionally well-isolated and carefully pre-pared microscopic and mesoscopic systems, the interac-tion of the system with the environment causes the off-diagonal terms of the local density matrix, expressed inthe pointer basis and describing the probability distribu-tion of the possible outcomes of a measurement on thesystem, to become extremely small in a very short pe-riod of time, and this process is irreversible for all prac-tical purposes.

E. Environment-induced superselection

Let us now turn to the second main consequence ofthe interaction with the environment, namely, theenvironment-induced selection of stable preferred-basisstates. We discussed in Sec. II.C the fact that thequantum-mechanical measurement scheme as repre-sented by Eq. s2.1d does not uniquely define the expan-sion of the postmeasurement state and thereby leavesopen the question of which observable can be consid-ered as having been measured by the apparatus. Thissituation is changed by the inclusion of the environmentstates in Eq. s3.5d for the following two reasons:

s1d Environment-induced superselection of a preferredbasis. The interaction between the apparatus andthe environment singles out a set of mutually com-muting observables.

s2d The existence of a tridecompositional uniquenesstheorem sElby and Bub, 1994; Clifton, 1995; Bub,1997d. If a state ucl in a Hilbert space H1 ^ H2 ^ H3can be decomposed into the diagonal s“Schmidt”dform ucl=oiaiufil1ufil2ufil3, the expansion is uniqueprovided that the hufil1j and hufil2j are sets of lin-early independent, normalized vectors in H1 and H2,respectively, and that hufil3j is a set of mutually non-collinear normalized vectors in H3. This can be gen-eralized to an N-decompositional uniqueness theo-rem, in which Nù3. Note that it is not alwayspossible to decompose an arbitrary pure state ofmore than two systems sNù3d into the Schmidtform ucl=oiaiufil1ufil2¯ ufilN, but if the decompo-sition exists, its uniqueness is guaranteed.

The tridecompositional uniqueness theorem ensuresthat the expansion of the final state in Eq. s3.5d is

unique, which fixes the ambiguity in the choice of the setof possible outcomes. It demonstrates that the inclusionof sat leastd a third “system” shere referred to as theenvironmentd is necessary to remove the basis ambigu-ity.

Of course, given any pure state in the composite Hil-bert space H1 ^ H2 ^ H3, the tridecompositional unique-ness theorem neither tells us whether a Schmidt decom-position exists nor specifies the unique expansion itselfsprovided the decomposition is possibled, and since theprecise states of the environment are generally notknown, an additional criterion is needed that determineswhat the preferred states will be.

1. Stability criterion and pointer basis

The decoherence program has attempted to definesuch a criterion based on the interaction with the envi-ronment and the idea of robustness and preservation ofcorrelations. The environment thus plays a double rolein suggesting a solution to the preferred-basis problem:it selects a preferred pointer basis, and it guarantees itsuniqueness via the tridecompositional uniqueness theo-rem.

In order to motivate the basis superselection approachproposed by the decoherence program, we note that instep s2d of Eq. s3.5d we tacitly assumed that interactionwith the environment does not disturb the establishedcorrelation between the state of the system, usnl, and thecorresponding pointer state uanl. This assumption can beviewed as a generalization of the concept of “faithfulmeasurement” to the realistic case in which the environ-ment is included. Faithful measurement in the usualsense concerns step s1d, namely, the requirement that themeasuring apparatus A act as a reliable “mirror” of thestates of the system S by forming only correlations ofthe form usnluanl but not usmluanl with mÞn. But sinceany realistic measurement process must include the in-evitable coupling of the apparatus to its environment,the measurement could hardly be considered faithful asa whole if the interaction with the environment dis-turbed the correlations between the system and theapparatus.9

It was therefore first suggested by Zurek s1981d thatthe preferred pointer basis be taken as the basis that“contains a reliable record of the state of the system S”sZurek, 1981, p. 1519d, i.e., the basis in which the system-apparatus correlations usnluanl are left undisturbed bythe subsequent formation of correlations with the envi-ronment sthe stability criteriond. One can then find a suf-ficient criterion for dynamically stable pointer states thatpreserve the system-apparatus correlations in spite ofthe interaction of the apparatus with the environment by

requiring all pointer state projection operators PnsAd

9For fundamental limitations on the precision of von Neu-mann measurements of operators that do not commute with aglobally conserved quantity, see the Wigner-Araki-Yanasetheorem sWigner, 1952; Araki and Yanase, 1960d.



= uanlkanu to commute with the apparatus-environment

interaction Hamiltonian HAE,10

fPnsAd,HAEg = 0 for all n . s3.21d

This implies that any correlation of the measured systemsor any other system, for instance, an observerd with theeigenstates of a preferred apparatus observable,

OA = on

lnPnsAd, s3.22d

is preserved, and that the states of the environment re-

liably mirror the pointer states PnsAd. In this case, the

environment can be regarded as carrying out a non-demolition measurement on the apparatus. The commu-tativity requirement, Eq. s3.21d, is obviously fulfilled if

HAE is a function of OA ,HAE=HAEsOAd. Conversely,system-apparatus correlations in which the states of theapparatus are not eigenstates of an observable that com-

mutes with HAE will, in general, be rapidly destroyed bythe interaction.

Put the other way around, this implies that the envi-ronment determines, through the form of the interaction

Hamiltonian HAE, a preferred apparatus observable OA,Eq. s3.22d, and thereby also the states of the system thatare measured by the apparatus, that is, reliably recordedthrough the formation of dynamically stable quantumcorrelations. The tridecompositional uniqueness theo-rem then guarantees the uniqueness of the expansion ofthe final state ucl=oncnusnluanluenl swhere no constraintson the cn have to be imposedd and thereby the unique-ness of the preferred pointer basis.

Other criteria similar to the commutativity require-ment, Eq. s3.21d, have been suggested for the selectionof the preferred pointer basis because it turns out that inrealistic cases the simple relation of Eq. s3.21d can usu-ally only be fulfilled approximately sZurek, 1993; Zureket al., 1993d. More general criteria, for example,have been based on the von Neumann entropy−Tr rC

2 stdln rC2 std, or the purity Tr rC

2 std, with the goal offinding the most robust states or the states which be-come least entangled with the environment in the courseof the evolution sZurek, 1993, 1998, 2003a; Zurek et al.,1993d. Pointer states are obtained by extremizing themeasure si.e., minimizing entropy, or maximizing purity,etc.d over the initial state uCl and requiring the resultingstates to be robust when varying the time t. Applicationof this method leads to a ranking of the possible pointerstates with respect to their “classicality,” i.e., their ro-bustness with respect to interaction with the environ-ment, and thus allows for the selection of a preferredpointer basis in terms of the “most classical” pointerstates sthe predictability sieve; see Zurek, 1993; Zurek etal., 1993d. Although the proposed criteria differ some-

what and other meaningful criteria are likely to be sug-gested in the future, it is hoped that in the macrosopiclimit the resulting stable pointer states obtained fromdifferent criteria will turn out to be very similar sZurek,2003ad. For some toy models sin particular, forharmonic-oscillator models that lead to coherent statesas pointer statesd, this has already been verified explic-itly ssee, for example, Kübler and Zeh, 1973; Zurek etal., 1993; Diósi and Kiefer, 2000; Joos et al., 2003; Eisert,2004d.

2. Selection of quasiclassical properties

System-environment interaction Hamiltonians fre-quently describe a scattering process of surrounding par-ticles sphotons, air molecules, etc.d interacting with thesystem under study. Since the force laws describing suchprocesses typically depend on some power of distancessuch as ~r−2 in Newton’s or Coulomb’s force lawd, theinteraction Hamiltonian will usually commute with theposition basis, such that, according to the commutativityrequirement of Eq. s3.21d, the preferred basis will be inposition space. The fact that position is frequently thedeterminate property of our experience can then be ex-plained by referring to the dependence of most interac-tions on distance sZurek, 1981, 1982, 1991d.

This holds, in particular, for mesoscopic and macro-scopic systems, as demonstrated, for instance, by thepioneering study of Joos and Zeh s1985d, in which sur-rounding photons and air molecules are shown to con-tinuously “measure” the spatial structure of dust par-ticles, leading to rapid decoherence into an apparentsimproperd mixture of wave packets that are sharplypeaked in position space. Similar results sometimes evenhold for microscopic systems susually found in energyeigenstates; see belowd when they occur in distinct spa-tial structures that couple strongly to the surroundingmedium. For instance, chiral molecules such as sugar arealways observed to be in chirality eigenstates sleft-handed and right-handedd which are superpositions ofdifferent energy eigenstates sHarris and Stodolsky, 1981;Zeh, 2000d. This is explained by the fact that the spatialstructure of these molecules is continuously “moni-tored” by the environment, for example, through thescattering of air molecules, which gives rise to a muchstronger coupling than could typically be achieved by ameasuring device that was intended to measure, say, par-ity or energy; furthermore, any attempt to prepare suchmolecules in energy eigenstates would lead to immedi-ate decoherence into environmentally stable s“dynami-cally robust”d chirality eigenstates, thus selecting posi-tion as the preferred basis.

On the other hand, it is well known that many sys-tems, especially in the microsopic domain, are typicallyfound in energy eigenstates, even if the interactionHamiltonian depends on a different observable than en-ergy, e.g., position. Paz and Zurek s1999d have shownthat this situation arises when the predominant frequen-cies present in the environment are significantly lowerthan the intrinsic frequencies of the system, that is, when

10For simplicity, we assume here that the environment E in-teracts directly only with the apparatus A, but not with thesystem S.



the separation between the energy states of the system isgreater than the largest energies available in the envi-ronment. Then, the environment will only be able tomonitor quantities that are constants of motion. In thecase of nondegeneracy, this will be energy, thus leadingto the environment-induced superselection of energyeigenstates for the system.

Another example of environment-induced superselec-tion that has been studied is related to the fact that onlyeigenstates of the charge operator are observed, butnever superpositions of different charges. The existenceof the corresponding superselection rules was first onlypostulated sWick et al., 1952, 1970d, but could subse-quently be explained in the framework of decoherenceby referring to the interaction of the charge with its ownCoulomb sfard field, which takes the role of an “environ-ment,” leading to immediate decoherence of charge su-perpositions into an apparent mixture of charge eigen-states sGiulini et al., 1995; Giulini, 2000d.

In general, three different cases have typically beendistinguished sfor example, in Paz and Zurek, 1999d forthe kind of pointer observable emerging from an inter-action with the environment, depending on the relative

strengths of the system’s self-Hamiltonian HS and of the

system-environment interaction Hamiltonian HSE:

s1d When the dynamics of the system are dominated by

HSE, i.e., the interaction with the environment, the

pointer states will be eigenstates of HSE sand thustypically eigenstates of positiond. This case corre-sponds to the typical quantum measurement setting;see, for example, the model of Zurek s1981, 1982d,which is outlined in Sec. III.D.2 above.

s2d When the interaction with the environment is weak

and HS dominates the evolution of the system sthatis, when the environment is “slow” in the abovesensed, a case that frequently occurs in the micro-scopic domain, pointer states will arise that are en-

ergy eigenstates of HS sPaz and Zurek, 1999d.

s3d In the intermediate case, when the evolution of the

system is governed by HSE and HS in roughly equalstrength, the resulting preferred states will representa “compromise” between the first two cases; for in-stance, the frequently studied model of quantumBrownian motion has shown the emergence ofpointer states localized in phase space, i.e., in bothposition and momentum sUnruh and Zurek, 1989;Zurek et al., 1993; Joos et al., 2003; Zurek, 2003a;Eisert, 2004d.

3. Implications for the preferred-basis problem

The decoherence program proposes that the preferredbasis be selected by the requirement that correlations bepreserved in spite of the interaction with the environ-ment, and thus be chosen through the form of thesystem-environment interaction Hamiltonian. Thisseems certainly reasonable, since only such “robust”

states will in general be observable—and, after all, weseek only an explanation for our experience ssee thediscussion in Sec. II.B.3d. Although only particular ex-amples have been studied sfor a survey and references,see, for example, Blanchard et al., 2000; Joos et al., 2003;Zurek, 2003ad, the results thus far suggest that the se-lected properties are in agreement with our observation:for mesoscopic and macroscopic objects the distance-dependent scattering interaction with surrounding airmolecules, photons, etc., will in general give rise to im-mediate decoherence into spatially localized wave pack-ets and thus select position as the preferred basis. Onthe other hand, when the environment is comparably“slow,” as is frequently the case for microsopic systems,environment-induced superselection will typically yieldenergy eigenstates as the preferred states.

The clear merit of the approach of environment-induced superselection lies in the fact that the preferredbasis is not chosen in an ad hoc manner simply to makeour measurement records determinate or to match ourexperience of which physical quantities are usually per-ceived as determinate sfor example, positiond. Insteadthe selection is motivated on physical, observer-freegrounds, that is, through the system-environment inter-action Hamiltonian. The vast space of possiblequantum-mechanical superpositions is reduced so muchbecause the laws governing physical interactions dependonly on a few physical quantities sposition, momentum,charge, and the liked, and the fact that precisely theseare the properties that appear determinate to us is ex-plained by the dependence of the preferred basis on theform of the interaction. The appearance of “classicality”is therefore grounded in the structure of the physicallaws—certainly a highly satisfying and reasonable ap-proach.

The above argument in favor of the approach ofenvironment-induced superselection could, of course, beconsidered as inadequate on a fundamental level: Allphysical laws are discovered and formulated by us, sothey can contain only the determinate quantities of ourexperience. These are the only quantities we can per-ceive and thus include in a physical law. Thus the deri-vation of determinacy from the structure of our physicallaws might seem circular. However, we argue again thatit suffices to demand a subjective solution to thepreferred-basis problem—that is, to provide an answerto the question of why we perceive only such a smallsubset of properties as determinate, not whether therereally are determinate properties son an ontologicalleveld and what they are scf. the remarks in Sec. II.B.3d.

We might also worry about the generality of this ap-proach. One would need to show that any suchenvironment-induced superselection leads, in fact, toprecisely those properties that appear determinate to us.But this would require precise knowledge of the systemand the interaction Hamiltonian. For simple toy models,the relevant Hamiltonians can be written down explic-itly. In more complicated and realistic cases, this will ingeneral be very difficult, if not impossible, since the formof the Hamiltonian will depend on the particular system



or apparatus and the monitoring environment underconsideration, where, in addition, the environment is notonly difficult to define precisely, but also constantlychanging, uncontrollable, and, in essence, infinitelylarge.

But the situation is not as hopeless as it might sound,since we know that the interaction Hamiltonian will, ingeneral, be based on the set of known physical lawswhich, in turn, employ only a relatively small number ofphysical quantities. So as long as we assume the stabilitycriterion and consider the set of known physical quanti-ties as complete, we can automatically anticipate thatthe preferred basis will be a member of this set. Theremaining, yet very relevant, question is then which sub-set of these properties will be chosen in a specific physi-cal situation sfor example, will the system preferably befound in an eigenstate of energy or of position?d, and towhat extent this will match the experimental evidence.To give an answer, one usually will need a more detailedknowledge of the interaction Hamiltonian and of itsrelative strength with respect to the self-Hamiltonian ofthe system in order to verify the approach. Besides, asmentioned in Sec. III.E, there exist other criteria thanthe commutativity requirement, and whether they alllead to the same determinate properties is a questionthat has not yet been fully explored.

Finally, a fundamental conceptual difficulty of thedecoherence-based approach to the preferred-basisproblem is the lack of a general criterion for what de-fines the systems and the “unobserved” degrees of free-dom of the environment ssee the discussion in Sec.III.Ad. While in many laboratory-type situations, the di-vision into system and environment might seem straight-forward, it is not clear a priori how quasiclassical observ-ables can be defined through environment-inducedsuperselection on a larger and more general scale, whenlarger parts of the universe are considered where thesplit into subsystems is not suggested by some specificsystem-apparatus-surroundings setup.

To summarize, environment-induced superselection ofa preferred basis sid proposes an explanation for why aparticular pointer basis gets chosen at all—by arguingthat it is only the pointer basis that leads to stable, andthus perceivable, records when the interaction of the ap-paratus with the environment is taken into account; andsiid argues that the preferred basis will correspond to asubset of the set of the determinate properties of ourexperience, since the governing interaction Hamiltonianwill depend solely on these quantities. But it does nottell us precisely what the pointer basis will be in anygiven physical situation, since it will usually be hardlypossible to write down explicitly the relevant interactionHamiltonian in realistic cases. This also means that itwill be difficult to argue that any proposed criterionbased on the interaction with the environment will al-ways and in all generality lead to exactly those proper-ties that we perceive as determinate.

More work remains to be done, therefore, to fully ex-plore the general validity and applicability of the ap-proach of environment-induced superselection. But

since the results obtained thus far from toy models havebeen in promising agreement with empirical data, thereis little reason to doubt that the decoherence programhas proposed a very valuable criterion for explaining theemergence of preferred states and their robustness. Thefact that the approach is derived from physical principlesshould be counted additionally in its favor.

4. Pointer basis vs instantaneous Schmidt states

The so-called Schmidt basis, obtained by diagonalizingthe sreducedd density matrix of the system at each in-stant of time, has been frequently studied with respect toits ability to yield a preferred basis ssee, for example,Zeh, 1973; Albrecht, 1992, 1993d, having led some toconsider the Schmidt-basis states as describing “instan-taneous pointer states” sAlbrecht, 1992d. However, as ithas been emphasized sfor example, by Zurek, 1993d, anydensity matrix is diagonal in some basis, and this basiswill in general not play any special interpretive role.Pointer states that are supposed to correspond to quasi-classical stable observables must be derived from an ex-plicit criterion for classicality stypically, the stability cri-teriond; the simple mathematical diagonalizationprocedure of the instantaneous density matrix will gen-erally not suffice to determine a quasiclassical pointerbasis ssee the studies by Barvinsky and Kamenshchik,1995; Kent and McElwaine, 1997d.

In a more refined method, one refrains from comput-ing instantaneous Schmidt states and instead allows for acharacteristic decoherence time tD to pass, during whichthe reduced density matrix decoheres sa process that canbe described by an appropriate master equationd andbecomes approximately diagonal in the stable pointerbasis, the basis that is selected by the stability criterion.Schmidt states are then calculated by diagonalizing thedecohered density matrix. Since decoherence usuallyleads to rapid diagonality of the reduced density matrixin the stability-selected pointer basis to a very good ap-proximation, the resulting Schmidt states are typicallyvery similar to the pointer basis except when the pointerstates are very nearly degenerate. The latter situation isreadily illustrated by considering the approximately di-agonalized decohered density matrix

r = S1/2 + d v*

v 1/2 − dD , s3.23d

where uvu!1 sstrong decoherenced and d!1 snear-degeneracy; Albrecht, 1993d. If decoherence led to exactdiagonality, v=0, the eigenstates would be, for any fixedvalue of d, proportional to s0,1d and s1,0d scorrespondingto the “ideal” pointer statesd. However, for fixed v.0sapproximate diagonalityd and d→0 sdegeneracyd, theeigenstates become proportional to s±uvu /v, 1d, whichimplies that, in the case of degeneracy, the Schmidt de-composition of the reduced density matrix can yield pre-ferred states that are very different from the stablepointer states, even if the decohered, rather than theinstantaneous, reduced density matrix is diagonalized.



In summary, it is important to emphasize that stabilitysor a similar criteriond is the relevant requirement forthe emergence of a preferred quasiclassical basis, whichcannot, in general, be achieved by simply diagonalizingthe instantaneous reduced density matrix. However, theeigenstates of the decohered reduced density matrixwill, in many situations, approximate the quasiclassicalstable pointer states well, especially when these pointerstates are sufficiently nondegenerate.

F. Envariance, quantum probabilities, and the Born rule

In the following, we shall review an interesting andpromising approach introduced recently by Zureks2003a, 2003b, 2004a, 2004bd that aims to explain theemergence of quantum probabilities and to deduce theBorn rule based on a mechanism termed “environment-assisted invariance,” or “envariance” for short, a par-ticular symmetry property of entangled quantum states.The original exposition of Zurek s2003ad was followedup by several articles by other authors, who assessed theapproach, pointed out more clearly the assumptions en-tering into the derivation, and presented variants of theproof sBarnum, 2003; Schlosshauer and Fine, 2003;Mohrhoff, 2004d. An expanded treatment of envarianceand quantum probabilities that addresses some of theissues discussed in these papers and that offers an inter-esting outlook on further implications of envariance canbe found in Zurek s2004ad. In our outline of the theoryof envariance, we shall follow this most recent treat-ment, as it spells out the derivation and the requiredassumptions more explicitly and in greater detail andclarity than in Zurek’s earlier s2003a, 2003b, 2004bd pa-pers scf. also the remarks of Schlosshauer and Fine,2003d.

We include a discussion of Zurek’s proposal here fortwo reasons. First, the derivation is based on the inclu-sion of an environment E, entangled with the system S ofinterest to which probabilities of measurement out-comes are to be assigned, and thus it matches well thespirit of the decoherence program. Second, and moreimportantly, despite the contributions of decoherence toexplaining the emergence of subjective classicality fromquantum mechanics, a consistent derivation of classical-ity sincluding a motivation for some of the axioms ofquantum mechanics, as suggested by Zurek, 2003ad re-quires the separate derivation of the Born rule. The de-coherence program relies heavily on the concept of re-duced density matrices and the related formalism andinterpretation of the trace operation, see Eq. s3.6d,which presuppose Born’s rule. Therefore decoherence it-self cannot be used to derive the Born rule sas was tried,for example, by Zurek, 1998, and Deutsch, 1999d sinceotherwise the argument would be rendered circularsZeh, 1997; Zurek, 2003ad.

There have been various attempts in the past to re-place the postulate of the Born rule by a derivation.Gleason’s s1957d theorem has shown that if one imposesthe condition that for any orthonormal basis of a givenHilbert space the sum of the probabilities associated

with each basis vector must add up to one, the Born ruleis the only possibility for the calculation of probabilities.However, Gleason’s proof provides little insight into thephysical meaning of the Born probabilities, and there-fore various other attempts, typically based on arelative-frequencies approach si.e., on a counting argu-mentd, have been made towards a derivation of the Bornrule in a no-collapse sand usually relative-stated settingssee, for example, Everett, 1957; Hartle, 1968; DeWitt,1971; DeWitt and Graham, 1973; Graham, 1973; Geroch,1984; Farhi et al., 1989; Deutsch, 1999d. However, it waspointed out that these approaches fail due to the use ofcircular arguments sStein, 1984; Kent, 1990; Squires,1990; Barnum et al., 2000d; cf. also Wallace s2003bd andSaunders s2002d.

Zurek’s recently developed theory of envariance pro-vides a promising new strategy for deriving, given cer-tain assumptions, the Born rule in a manner that avoidsthe circularities of the earlier approaches. We shall out-line the concept of envariance in the following and showhow it can lead to Born’s rule.

1. Environment-assisted invariance

Zurek introduces his definition of envariance as fol-lows: Consider a composite state ucSEl swhere, as usual,S refers to the “system” and E to some “environment”din a Hilbert space given by the tensor product HS ^ HE,

and a pair of unitary transformations US= uS ^ IE and

UE= IS ^ uE acting on S and E, respectively. If ucSEl is

invariant under the combined application of US and UE,

UEsUSucSEld = ucSEl , s3.24d

ucSEl is called envariant under uS. In other words, the

change in ucSEl induced by acting on S via US can be

undone by acting on E via UE. Note that envariance is adistinctly quantum feature, absent from pure classicalstates, and a consequence of quantum entanglement.

The main argument of Zurek’s derivation is based ona study of a composite pure state in the diagonalSchmidt decomposition

ucSEl =1Î2

seiw1us1lue1l + eiw2us1lue1ld , s3.25d

where the husklj and hueklj are sets of orthonormal basisvectors that span the Hilbert spaces HS and HE, respec-tively. The case of higher-dimensional state spaces canbe treated similarly, and a generalization to expansioncoefficients of different magnitudes can be made by ap-plication of a standard counting argument sZurek,2003b, 2004ad. The Schmidt states uskl are identified withthe outcomes, or “events” sZurek, 2004b, p. 12d, towhich probabilities are to be assigned.

Zurek now states three simple assumptions, called“facts” sZurek, 2004a, p. 4; see also the discussion inSchlosshauer and Fine, 2003d:



sA1d A unitary transformation of the form ¯^ IS ^¯

does not alter the state of S.

sA2d All measurable properties of S, including prob-abilities of outcomes of measurements on S, arefully determined by the state of S.

sA3d The state of S is completely specified by the glo-bal composite state vector ucSEl.

Given these assumptions, one can show that the stateof S and any measurable properties of S cannot be af-fected by envariant transformations. The proof goes asfollows. The effect of an envariant transformation uS^ IE acting on ucSEl can be undone by a corresponding

“countertransformation” IS ^ uE that restores the origi-nal state vector ucSEl. Since it follows from sA1d that thelatter transformation has left the state of S unchanged,but sA3d implies that the final state of S safter the trans-formation and countertransformationd is identical to the

initial state of S, the first transformation uS ^ IE cannothave altered the state of S either. Thus, using assump-tion sA2d, it follows that an envariant transformation

uS ^ IE acting on ucSEl leaves any measurable propertiesof S unchanged, in particular the probabilities associatedwith outcomes of measurements performed on S.

Let us now consider two different envariant transfor-mations: A phase transformation of the form

uSsj1,j2d = eij1us1lks1u + eij2us2lks2u s3.26d

that changes the phases associated with the Schmidtproduct states uskluekl in Eq. s3.25d, and a swap transfor-mation

uSs1 ↔ 2d = eij12us1lks2u + eij21us2lks1u s3.27d

that exchanges the pairing of the uskl with the uell. Basedon the assumptions sA1d–sA3d mentioned above, envari-ance of ucSEl under these transformations means thatmeasurable properties of S cannot depend on the phaseswk in the Schmidt expansion of ucSEl, Eq. s3.25d. Similar-ily, it follows that a swap uSs1↔2d leaves the state of Sunchanged, and that the consequences of the swap can-not be detected by any measurement that pertains to Salone.

2. Deducing the Born rule

Together with an additional assumption, this resultcan then be used to show that the probabilities of the“outcomes” uskl appearing in the Schmidt decompositionof ucSEl must be equal, thus arriving at Born’s rule forthe special case of a state-vector expansion with coeffi-cients of equal magnitude. Zurek s2004ad offers threepossibilities for such an assumption. Here we shall limitour discussion to one of these possible assumptions sseealso the comments in Schlosshauer and Fine, 2003d:

sA4d The Schmidt product states uskluekl appearing inthe state-vector expansion of ucSEl imply a directand perfect correlation of the measurement out-

comes associated with the uskl and uekl. That is, if

an observable OS=olslusllkslu is measured on S anduskl is obtained, a subsequent measurement of

OE=oleluellkelu on E will yield uekl with certaintysi.e., with probability equal to oned.

This assumption explicitly introduces a probabilityconcept into the derivation. sSimilarly, the two otherpossible assumptions suggested by Zurek establish aconnection between the state of S and probabilities ofoutcomes of measurements on S.d

Then, denoting the probability for the outcome uskl bypsuskl , ucSEld when the composite system SE is describedby the state vector ucSEl, this assumption implies that

psuskl ; ucSEld = psuekl ; ucSEld . s3.28d

After acting with the envariant swap transformation

US= uSs1↔2d ^ IE fsee Eq. s3.27dg on ucSEl and using as-sumption sA4d again, we get

psus1l ;USucSEld = psue2l ;USucSEld ,

psus2l ;USucSEld = psue1l ;USucSEld . s3.29d

Now, when a “counterswap” UE= IS ^ uEs1↔2d is ap-plied to ucSEl, the original state vector ucSEl is restored,

i.e., UEsUSucSEld= ucSEl. It then follows from assumptionssA2d and sA3d listed above that

psuskl ;UEUSucSEld = psuskl ; ucSEld . s3.30d

Furthermore, assumptions sA1d and sA2d imply that thefirst and second swaps cannot have affected the measur-able properties of E and S, respectively, particularly notthe probabilities for outcomes of measurements onE sSd,

psuskl ;UEUSucSEld = psuskl ;USucSEld ,

psuekl ;USucSEld = psuekl ; ucSEld . s3.31d

Combining Eqs. s3.28d–s3.31d yields

psus1l ; ucSEld =s3.30d

psus1l ;UEUSucSEld =s3.31d

psus1l ;USucSElds3.32d

=s3.29d

psue2l ;USucSEld =s3.31d

psue2l ; ucSEld

=s3.28d

psus2l ; ucSEld , s3.33d

which establishes the desired result psus1l ; ucSEld=psus2l ; ucSEld. The general case of unequal coefficientsin the Schmidt decomposition of ucSEl can then betreated by means of a simple counting method sZurek,2003b, 2004ad, leading to Born’s rule for probabilitiesthat are rational numbers. Using a continuity argument,this result can be further generalized to include prob-abilities that cannot be expressed as rational numberssZurek, 2004ad.



3. Summary and outlook

If one grants the stated assumptions, Zurek’s develop-ment of the theory of envariance offers a novel andpromising way of deducing Born’s rule in a noncircularmanner. Compared to the relatively well-studied field ofdecoherence, envariance and its consequences have onlybegun to be explored. In this review, we have focused onenvariance in the context of a derivation of the Bornrule, but other far-reaching implications of envariancehave recently been suggested by Zurek s2004ad. For ex-ample, envariance could also account for the emergenceof an environment-selected preferred basis sthat is, forenvironment-induced superselectiond without an appealto the trace operation or to reduced density matrices.This could open up the possibility of a redevelopment ofthe decoherence program based on fundamentalquantum-mechanical principles that do not require oneto presuppose the Born rule; this also might shed newlight, for example, on the interpretation of reduced den-sity matrices that has led to much controversy in discus-sions of decoherence ssee Sec. III.Bd. As of now, thedevelopment of such ideas is at a very early stage, butwe can expect further interesting results derived fromenvariance in the near future.

IV. THE ROLE OF DECOHERENCE IN INTERPRETATIONSOF QUANTUM MECHANICS

It was not until the early 1970s that the importance ofthe interaction of physical systems with their environ-ments for a realistic quantum-mechanical description ofthese systems was realized and a proper viewpoint onsuch interactions was established sZeh, 1970, 1973d. Ittook another decade for the first concise formulation ofthe theory of decoherence sZurek, 1981, 1982d to beworked out and for numerical studies to be made thatshowed the ubiquity and effectiveness of decoherenceeffects sJoos and Zeh, 1985d. Of course, by that time,several interpretive approaches to quantum mechanicshad already been established, for example, Everett-stylerelative-state interpretations sEverett, 1957d, the conceptof modal interpretations introduced by van Fraassens1973, 1991d, and the pilot-wave theory of de Broglie andBohm sBohm, 1952d.

When the relevance of decoherence effects was recog-nized by sparts ofd the scientific community, decoherenceprovided a motivation for a fresh look at the existinginterpretations and for the introduction of changes andextensions to these interpretations, as well as for newinterpretations. Some of the central questions in thiscontext were, and still are, the following:

s1d Can decoherence by itself solve certain foundationalissues at least for all practical purposes, such as tomake certain interpretive additives superfluous?What then, are, the crucial remaining foundationalproblems?

s2d Can decoherence protect an interpretation from em-pirical disproof?

s3d Conversely, can decoherence provide a mechanismto exclude an interpretive strategy as incompatiblewith quantum mechanics and/or as empirically inad-equate?

s4d Can decoherence physically motivate some of theassumptions on which an interpretation is based andgive them a more precise meaning?

s5d Can decoherence serve as an amalgam that wouldunify and simplify a spectrum of differentinterpretations?

These and other questions have been widely dis-cussed, both in the context of particular interpretationsand with respect to the general implications of decoher-ence for any interpretation of quantum mechanics. Inparticular, interpretations that uphold the universal va-lidity of the unitary Schrödinger time evolution, mostnotably relative-state and modal interpretations, havefrequently incorporated environment-induced superse-lection of a preferred basis and decoherence into theirframework. It is the purpose of this section to criticallyinvestigate the implications of decoherence for the exist-ing interpretations of quantum mechanics, with particu-lar attention to the questions outlined above.

A. General implications of decoherence for interpretations

When measurements are understood as ubiquitous in-teractions that lead to the formation of quantum corre-lations, the selection of a preferred basis becomes, inmost cases, a fundamental requirement. This also corre-sponds, in general, to the question of what propertiesare being ascribed to systems sor worlds, minds, etc.d.Thus the preferred-basis problem is at the heart of anyinterpretation of quantum mechanics. Some of the diffi-culties that must be faced in solving the preferred-basisproblem are

sid to decide whether the selection of any preferredbasis sor quantity or propertyd is justified at all oronly an artifact of our subjective experience;

siid if we decide on sid in the positive, to select thosedeterminate quantity or quantities swhat appearsdeterminate to us does not need to be appear de-terminate to other kinds of observers, nor does itneed to be the “true” determinate propertyd;

siiid to avoid any ad hoc character of the choice andany possible empirical inadequacy or inconsis-tency with the confirmed predictions of quantummechanics;

sivd if a multitude of quantities is selected that applydifferently among different systems, to be able toformulate explicit rules that specify the determi-nate quantity or quantities under every circum-stance;

svd to ensure that the basis is chosen such that if thesystem is embedded in a larger scomposited sys-tem, the principle of property composition holds,



i.e., the property selected by the basis of the origi-nal system should also persist when the system isconsidered as part of a larger composite system.11

The hope is then that environment-induced superselec-tion of a preferred basis can provide a universal mecha-nism that fulfills the above criteria and solves thepreferred-basis problem on strictly physical grounds.

A popular reading of the decoherence program typi-cally goes as follows. First, the interaction of the systemwith the environment selects a preferred basis, i.e., aparticular set of quasiclassical robust states that com-mute, at least approximately, with the Hamiltonian gov-erning the system-environment interaction. Since theform of the interaction Hamiltonians usually depends onfamiliar “classical” quantities, the preferred states willtypically also correspond to the small set of “classical”properties. Decoherence then quickly damps superposi-tions between the localized preferred states when onlythe system is considered. This is taken as an explanationof the appearance to a local observer of a “classical”world of determinate, “objective” sin the sense of beingrobustd properties. The tempting interpretation of theseachievements is then to conclude that this accounts forthe observation of unique svia environment-induced su-perselectiond and definite svia decoherenced pointerstates at the end of the measurement, and the measure-ment problem appears to be solved, at least for all prac-tical purposes.

However, the crucial difficulty in the above reasoningis justifying the second step: How is one to interpret thelocal suppression of interference in spite of the fact thatfull coherence is retained in the total state that describesthe system-environment combination? While the localdestruction of interference allows one to infer the emer-gence of an simproperd ensemble of individually local-ized components of the wave function, one still needs toimpose an interpretive framework that explains whyonly one of the localized states is realized and/or per-ceived. This has been done in various interpretations ofquantum mechanics, typically on the basis of the deco-hered reduced density matrix to ensure consistency withthe predictions of the Schrödinger dynamics and thus toguarantee empirical adequacy.

In this context, one might raise the question whetherretention of full coherence in the composite state of thesystem-environment combination could ever lead to em-pirical conflicts with the ascription of definite values tosmesoscopic and macroscopicd systems in somedecoherence-based interpretive approach. After all, onecould think of enlarging the system so as to include theenvironment in such a way that measurements couldnow actually reveal the persisting quantum coherenceeven on a macroscopic level. However, Zurek s1982d as-serted that such measurements would be impossible tocarry out in practice, a statement that was supported bya simple model calculation by Omnès s1992d for a body

with a macrosopic number s1024d of degrees of freedom.

B. The standard and the Copenhagen interpretations

As is well known, the standard interpretation s“ortho-dox” quantum mechanicsd postulates that every mea-surement induces a discontinuous break in the unitarytime evolution of the state through the collapse of thetotal wave function onto one of its terms in the state-vector expansion suniquely determined by the eigenba-sis of the measured observabled, which selects a singleterm in the superposition as representing the outcome.The nature of the collapse is not at all explained, andthus the definition of measurement remains unclear.Macroscopic superpositions are not a priori forbidden,but are never observed since any observation would en-tail a measurementlike interaction. In the following, weshall also consider a “Copenhagen” variant of the stan-dard interpretation, which adds an additional key ele-ment, postulating the necessity of classical concepts inorder to describe quantum phenomena, including mea-surements.

1. The problem of definite outcomes

The interpretive rule of orthodox quantum mechanicsthat tells us when we can speak of outcomes is given bythe e-e link.12 This is an “objective” criterion since itallows us to infer the existence of a definite state in thesystem to which a value of a physical quantity can beascribed. Within this interpretive framework sand with-out presuming the collapse postulated decoherence can-not solve the problem of outcomes: Phase coherence be-tween macroscopically different pointer states ispreserved in the state that includes the environment,and we can always enlarge the system so as to include satleast parts ofd the environment. In other words, the su-perposition of different pointer positions still exists; co-herence is only “delocalized into the larger system”sKiefer and Joos, 1999, p. 5d, that is, into theenvironment—or, as Joos and Zeh s1985, p. 224d put it,“the interference terms still exist, but they are notthere”—and the process of decoherence could, in prin-ciple, always be reversed. Therefore, if we assume theorthodox e-e link to establish the existence of determi-nate values of physical quantities, decoherence cannotensure that the measuring device actually ever is in adefinite pointer state sunless, of course, the system isinitially in an eigenstate of the observabled, or that mea-surements have outcomes at all. Much of the generalcriticism directed against decoherence with respect to itsability to solve the measurement problem sat least in thecontext of the standard interpretationd has been cen-tered on this argument.

11This is a problem encountered in some modal interpreta-tions ssee Clifton, 1996d.

12It is not particularly relevant for the subsequent discussionwhether the e-e link is assumed in its “exact” form, i.e., requir-ing the exact eigenstates of an observable, or a “fuzzy” formthat allows the assignment of definiteness based on only ap-proximate eigenstates or on wave functions with stinyd “tails.”



Note that, with respect to the global postmeasurementstate vector, given by the final step in Eq. s3.5d, the in-teraction with the environment has only led to addi-tional entanglement. It has not transformed the statevector in any way, since the rapidly increasing orthogo-nality of the states of the environment associated withthe different pointer positions has not influenced thestate description at all. In brief, the entanglementbrought about by interaction with the environmentcould even be considered as making the measurementproblem worse. Bacciagaluppi s2003a, Sec. 3.2d puts itlike this:

Intuitively, if the environment is carrying out, with-out our intervention, lots of approximate positionmeasurements, then the measurement problemought to apply more widely, also to these sponta-neously occurring measurements. s…d The state ofthe object and the environment could be a super-position of zillions of very well localised terms,each with slightly different positions, and whichare collectively spread over a macroscopic dis-tance, even in the case of everyday objects. s…d Ifeverything is in interaction with everything else,everything is entangled with everything else, andthat is a worse problem than the entanglement ofmeasuring apparatuses with the measured probes.

Only once we have formed the reduced pure-state den-sity matrix rSA, Eq. s3.8d, can the orthogonality of theenvironmental states have an effect; then, rSA dynami-cally evolves into the improper ensemble rSA

d fEq. s3.9dg.However, as pointed out in our general discussion ofreduced density matrices in Sec. III.B, the orthodox ruleof interpreting superpositions prohibits regarding thecomponents in the sum of Eq. s3.9d as corresponding toindividual well-defined quantum states.

Rather than considering the postdecoherence state ofthe system sor, more precisely, of the system-apparatuscombination SAd, we can instead analyze the influenceof decoherence on the expectation values of observablespertaining to SA; after all, such expectation values arewhat local observers would measure in order to arrive atconclusions about SA. The diagonalized reduced densitymatrix, Eq. s3.9d, together with the trace relation, Eq.s3.6d, implies that, for all practical purposes, the statisticsof the system SA will be indistinguishable from that of aproper mixture sensembled by any local observation on

SA. That is, given sid the trace rule kOl=TrsrOd and siidthe interpretation of kOl as the expectation value of an

observable O, the expectation value of any observable

OSA restricted to the local system SA will be, for allpractical purposes, identical to the expectation value ofthis observable if SA had been in one of the statesusnluanl sas if SA were described by an ensemble ofstatesd. In other words, decoherence has effectively re-moved any interference terms ssuch as usmluamlkanuksnuwhere mÞnd from the calculation of the trace

TrsrSAOSAd and thereby from the calculation of the ex-

pectation value kOSAl. It has therefore been claimedthat formal equivalence—i.e., the fact that decoherencetransforms the reduced density matrix into a form iden-tical to that of a density matrix representing an en-semble of pure states—yields observational equivalencein the sense above, namely, the slocald indistinguishabil-ity of the expectation values derived from these twotypes of density matrices via the trace rule.

But we must be careful in interpreting the correspon-dence between the mathematical formalism ssuch as thetrace ruled and the common terms employed in describ-ing “the world.” In quantum mechanics, the identifica-tion of the expression “TrsrAd” as the expectation valueof a quantity relies on the mathematical fact that, whenwriting out this trace, it is found to be equal to a sumover the possible outcomes of the measurement,weighted by the Born probabilities for the system to be“thrown” into a particular state corresponding to eachof these outcomes in the course of a measurement. Thiscertainly represents our common-sense intuition aboutthe meaning of expectation values as the sum over pos-sible values that can appear in a given measurement,multiplied by the relative frequency of actual occurrenceof these values in a series of such measurements. Thisinterpretation, however, presumes sid that measurementshave outcomes, siid that measurements lead to definite“values,” siiid that measurable physical quantities areidentified as operators sobservablesd in a Hilbert space,and sivd that the modulus square of the expansion coef-ficients of the state in terms of the eigenbasis of theobservable can be interpreted as representing probabili-ties of actual measurement outcomes sBorn ruled.

Thus decoherence brings about an apparent sand ap-proximated mixture of states that seem, based on themodels studied, to correspond well to those states thatwe perceive as determinate. Moreover, our observationtells us that this apparent mixture indeed looks like aproper ensemble in a measurement situation, as we ob-serve that measurements lead to the “realization” ofprecisely one state in the “ensemble.” But within theframework of the orthodox interpretation, decoherencecannot explain this crucial step from an apparent mix-ture to the existence and/or perception of single out-comes.

2. Observables, measurements, and environment-induced superselection

In the standard and Copenhagen interpretations,property ascription is determined by an observable thatrepresents the measurement of a physical quantity andthat in turn defines the preferred basis. However, anyHermitian operator can play the role of an observable,and thus any given state has the potential for an infinitenumber of different properties whose attribution is usu-ally mutually exclusive unless the corresponding observ-ables commute sin which case they share a commoneigenbasis, which preserves the uniqueness of the pre-ferred basisd. What then determines the observable thatis being measured? As our discussion in Sec. II.C has



demonstrated, the derivation of the measured observ-able from the particular form of a given state-vector ex-pansion can lead to paradoxical results since this expan-sion is in general nonunique, so the observable must bechosen by other means. In the standard and Copen-hagen interpretations, it is essentially the “user” who“chooses” the particular observable to be measured andthus determines which properties the system possesses.

This positivist point of view has, of course, led to a lotof controversy, since it runs counter to the attempt toestablish an observer-independent reality that has beenthe central pursuit of natural science since its beginning.Moreover, in practice, one certainly does not have thefreedom to choose any arbitrary observable and mea-sure it; instead, one has “instruments” sincluding one’ssensesd that are designed to measure a particular observ-able. For most sand maybe alld practical purposes, thiswill ultimately boil down to a single relevant observable,namely, position. But what, then, makes the instrumentsdesigned for such a particular observable?

Answering this crucial question essentially meansabandoning the orthodox view of treating measurementsas a “black box” process that has little, if any, relation tothe workings of actual physical measurements swheremeasurements can here be understood in the broadestsense of a “monitoring” of the state of the systemd. Thefirst key point, the formalization of measurements as aformation of quantum correlations between system andapparatus, goes back to the early years of quantum me-chanics and is reflected in the measurement scheme ofvon Neumann s1932d, but it does not resolve the issue ofhow the choice of observables is made. The second keypoint, the explicit inclusion of the environment in a de-scription of the measurement process, was brought intoquantum theory by the studies of decoherence. Zurek’ss1981d stability criterion discussed in Sec. III.E hasshown that measurements must be of such a nature as toestablish stable records, where stability is to be under-stood as preserving the system-apparatus correlations inspite of the inevitable interaction with the surroundingenvironment. The “user” cannot choose the observablesarbitrarily, but must design a measuring device whoseinteraction with the environment is such as to ensurestable records in the sense above swhich, in turn, definesa measuring device for this observabled. In the readingof orthodox quantum mechanics, this can be interpretedas the environment determining the properties of thesystem.

In this sense, the decoherence program has embeddedthe rather formal concept of measurement as proposedby the standard and Copenhagen interpretations—withits vague notion of observables that are seemingly freelychosen by the observer—in a more realistic and physicalframework. This is accomplished via the specification ofobserver-free criteria for the selection of the measuredobservable through the physical structure of the measur-ing device and its interaction with the environment,which is, in most cases, needed to amplify the measure-ment record and thereby to make it accessible to theexternal observer.

3. The concept of classicality in the Copenhageninterpretation

The Copenhagen interpretation additionally postu-lates that classicality is not to be derived from quantummechanics, for example, as the macroscopic limit of anunderlying quantum structure sas is in some sense as-sumed, but not explicitly derived, in the standard inter-pretationd, but instead that it be viewed as an indispens-able and irreducible element of a complete quantumtheory—and, in fact, be considered as a concept prior toquantum theory. In particular, the Copenhagen interpre-tation assumes the existence of macroscopic measure-ment apparatuses that obey classical physics and that arenot supposed to be described in quantum-mechanicalterms sin sharp contrast to the von Neumann measure-ment scheme, which rather belongs to the standard in-terpretationd; such a classical apparatus is considerednecessary in order to make quantum-mechanical phe-nomena accessible to us in terms of the “classical” worldof our experience. This strict dualism between the sys-tem S, to be described by quantum mechanics, and theapparatus A, obeying classical physics, also entails theexistence of an essentially fixed boundary between Sand A, which separates the microworld from the macro-world sthe “Heisenberg cut”d. This boundary cannot bemoved significantly without destroying the observedphenomenon si.e., the full interacting compound SAd.

Especially in the light of the insights gained from de-coherence, it seems impossible to uphold the notion of afixed quantum-classical boundary on a fundamentallevel of the theory. Environment-induced superselectionand suppression of interference have demonstrated howquasiclassical robust states can emerge, or remain ab-sent, using the quantum formalism alone and over abroad range of microscopic to macroscopic scales, andhave established the notion that the boundary betweenS and A is to a large extent movable towards A. Similarresults have been obtained from the general study ofquantum nondemolition measurements ssee, for ex-ample, Chap. 19 of Auletta, 2000d which include themonitoring of a system by its environment. Also notethat, since the apparatus is described in classical terms, itis macroscopic by definition; but not every apparatusmust be macroscopic: the actual “instrument” could wellbe microscopic. Only the “amplifier” must be macro-scopic. As an example, consider Zurek’s s1981d toymodel of decoherence, outlined in Sec. III.D.2, in whichthe instrument can be represented by a bistable atomwhile the environment plays the role of the amplifier; amore realistic example is a macrosopic detector of gravi-tational waves that is treated as a quantum-mechanicalharmonic oscillator.

Based on the progress already achieved by the deco-herence program, it is reasonable to anticipate that de-coherence embedded in some additional interpretivestructure could lead to a complete and consistent deri-vation of the classical world from quantum-mechanicalprinciples. This would make the assumption of an intrin-sically classical apparatus swhich has to be treated out-



side of the realm of quantum mechanicsd, appear as nei-ther a necessary nor a viable postulate. Bacciagaluppis2003b, p. 22d refers to this strategy as “having Bohr’scake and eating it”: acknowledging the correctness ofBohr’s notion of the necessity of a classical world s“hav-ing Bohr’s cake”d, but being able to view the classicalworld as part of and as emerging from a purelyquantum-mechanical world.

C. Relative-state interpretations

Everett’s original s1957d proposal of a relative-stateinterpretation of quantum mechanics has motivated sev-eral strands of interpretation, presumably owing to thefact that Everett himself never clearly spelled out howhis theory was supposed to work. The system-observerduality of orthodox quantum mechanics introduces intothe theory external “observers” who are not describedby the deterministic laws of quantum systems but in-stead follow a stochastic indeterminism. This approachobviously runs into problems when the universe as awhole is considered: by definition, there cannot be anyexternal observers. The central idea of Everett’s pro-posal is then to abandon duality and instead sid to as-sume the existence of a total state uCl representing thestate of the entire universe and siid to uphold the univer-sal validity of the Schrödinger evolution, while siiid pos-tulating that all terms in the superposition of the totalstate at the completion of the measurement actually cor-respond to physical states. Each such physical state canbe understood as relative sad to the state of the otherpart in the composite system sas in Everett’s originalproposal; also see, Rovelli, 1996; Mermin, 1998ad, sbd toa particular “branch” of a constantly “splitting” universesthe many-worlds interpretations, popularized by DeWitt, 1970 and Deutsch, 1985d, or scd to a particular“mind” in the set of minds of the conscious observersthe many-minds interpretation; see, for example, Lock-wood, 1996d. In other words, every term in the final-statesuperposition can be viewed as representing an equally“real” physical state of affairs that is realized in a differ-ent “branch of reality.”

Decoherence adherents have typically been inclinedtowards relative-state interpretations sfor instance, Zeh,1970, 1973, 1993; Zurek, 1998d, presumably because theEverett approach takes unitary quantum mechanics es-sentially “as is,” with a minimum of added interpretiveelements. This matches well the spirit of the decoher-ence program, which attempts to explain the emergenceof classicality purely from the formalism of basic quan-tum mechanics. It may also seem natural to identify thedecohering components of the wave function with differ-ent Everett branches. Conversely, proponents ofrelative-state interpretations have frequently employedthe mechanism of decoherence in solving the difficultiesassociated with this class of interpretations ssee, for ex-ample, Deutsch, 1985, 1996, 2002; Saunders, 1995, 1997,1998; Vaidman, 1998; Wallace, 2002, 2003ad.

There are many different readings and versions ofrelative-state interpretations, especially with respect to

what defines the “branches,” “worlds,” and “minds”—whether we deal with one, a multitude, or an infinity ofsuch worlds and minds; and whether there is an actualsphysicald or only perspectival splitting of the worlds andminds into different branches corresponding to theterms in the superposition. Does the world or mind splitinto two separate copies sthus somehow doubling all thematter contained in the orginal systemd, or is there just a“reassignment” of states to a multitude of worlds orminds of constant stypically infinited number, or is thereonly one physically existing world or mind in which eachbranch corresponds to different “aspects” swhateverthey ared. Regardless, in the following discussion of thekey implications of decoherence, the precise details anddifferences of these various strands of interpretationwill, for the most part, be largely irrelevant.

Relative-state interpretations face two core difficul-ties. First, the preferred-basis problem: If states are onlyrelative, the question arises, relative to what? What de-termines the particular basis terms that are used to de-fine the branches, which in turn define the worlds orminds in the next instant of time? When precisely doesthe “splitting” occur? Which properties are made deter-minate in each branch, and how are they connected tothe determinate properties of our experience? Second,what is the meaning of probabilities, since every out-come actually occurs in some world or mind, and howcan Born’s rule be motivated in such an interpretiveframework?

1. Everett branches and the preferred-basis problem

Stapp s2002, p. 1043d stated the requirement that “amany-worlds interpretation of quantum theory existsonly to the extent that the associated basis problem issolved.” In the context of relative-state interpretations,the preferred-basis problem is not only much more se-vere than in the orthodox interpretation, but also morefundamental for several reasons: sid The branching oc-curs continuously and essentially everywhere; in thegeneral sense of measurements understood as the for-mation of quantum correlations, every newly formedcorrelation, whether it pertains to microscopic or mac-roscopic systems, corresponds to a branching. siid Theontological implications are much more drastic, at leastin those relative-state interpretations which assume anactual “splitting” of worlds or minds, since the choice ofthe basis determines the resulting “world” or “mind” asa whole.

The environment-based basis superselection criteriaof the decoherence program have frequently been em-ployed to solve the preferred-basis problem of relative-state interpretations ssee, for example, Zurek, 1998; But-terfield, 2001; Wallace, 2002, 2003ad. There are severaladvantages in a decoherence-related approach to select-ing the preferred Everett bases: First, no a priori exis-tence of a preferred basis needs to be postulated, butinstead the preferred basis arises naturally from thephysical criterion of robustness. Second, the selectionwill be likely to yield empirical adequacy, since the de-



coherence program is derived solely from the well-confirmed Schrödinger dynamics smodulo the possibilitythat robustness may not be the universally valid crite-riond. Lastly, the decohered components of the wavefunction evolve in such a way that they can be reidenti-fied over time sforming “trajectories” in the preferredstate spacesd, and thus can be used to define stable, tem-porally extended Everett branches. Similarly, such tra-jectories can be used to ensure robust observer recordstates and/or environmental states that make informa-tion about the state of the system of interest widely ac-cessible to observers ssee, for example, Zurek’s “existen-tial interpretation,” outlined in Sec. IV.C.3 belowd.

While the idea of directly associating theenvironment-selected basis states with Everett worldsseems natural and straightforward, it has also been sub-ject to criticism. Stapp s2002d has argued that anEverett-type interpretation must aim at determining adenumerable set of distinct branches that correspond tothe apparently discrete events of our experience.Among these branches one must be able to assign deter-minate values and finite probabilities according to theusual rules and therefore one would need to be able tospecify a denumerable set of mutually orthogonal pro-jection operators. It is well known, however sZurek,1998d, that the preferred states chosen through the inter-action with the environment via the stability criterionfrequently form an overcomplete set of states—often acontinuum of narrow Gaussian-type wave packets, forexample, the coherent states of harmonic-oscillatormodels that are not necessarily orthogonal si.e., theGaussians may overlap; see Kübler and Zeh, 1973;Zurek et al., 1993d. Stapp therefore considers this ap-proach to the preferred-basis problem in relative-stateinterpretations to be unsatisfactory. Zurek s2003cd hasrebutted this criticism by pointing out that a collectionof harmonic oscillators that would lead to such overcom-plete sets of Gaussians cannot serve as an adequatemodel of the human brain, and it is ultimately only inthe brain where the perception of denumerability andmutual exclusiveness of events must be accounted forssee Sec. II.B.3d; when neurons are more appropriatelymodeled as two-state systems, the issue raised by Stappdisappears sfor a discussion of decoherence in a simpletwo-state model, see Sec. III.D.2d.13

The approach of using environment-induced superse-lection and decoherence to define the Everett brancheshas also been critized on grounds of being “conceptuallyapproximate,” since the stability criterion generallyleads only to an approximate specification of a preferredbasis and therefore cannot give an “exact” definition ofthe Everett branches ssee, for example, the comments ofZeh, 1973, Kent, 1990, and also the well-known “anti-FAPP” position of Bell, 1982d. Wallace s2003a, pp. 90

and 91d has argued against such an objection as

s…d arising from a view implicit in much discussionof Everett-style interpretations: that certain con-cepts and objects in quantum mechanics must ei-ther enter the theory formally in its axiomaticstructure, or be regarded as illusion. s…d fInsteadgthe emergence of a classical world from quantummechanics is to be understood in terms of theemergence from the theory of certain sorts ofstructures and patterns, and … this means that wehave no need sas well as no hope!d of the precisionwhich Kent fin his s1990d critiqueg and others s…ddemand.

Accordingly, in view of our argument in Sec. II.B.3 forconsidering subjective solutions to the measurementproblem as sufficient, there is no a priori reason to doubtthat an “approximate” criterion for the selection of thepreferred basis can give a meaningful definition of theEverett branches—one that is empirically adequate andthat accounts for our experiences. The environment-superselected basis emerges naturally from the physi-cally very reasonable criterion of robustness togetherwith the purely quantum-mechanical effect of decoher-ence. It would be rather difficult to imagine that an axi-omatically introduced “exact” rule could be able to se-lect preferred bases in a manner that is similarlyphysically motivated and capable of ensuring empiricaladequacy.

Besides using the environment-superselected pointerstates to describe the Everett branches, various authorshave directly used the instantaneous Schmidt decompo-sition of the composite state sor, equivalently, the set oforthogonal eigenstates of the reduced density matrixd todefine the preferred basis ssee also Sec. III.E.4d. Thisapproach is easier to implement than the explicit searchfor dynamically stable pointer states, since the preferredbasis follows directly from a simple mathematical diago-nalization procedure at each instant of time. Further-more, it has been favored by some se.g., Zeh, 1973d sinceit gives an “exact” rule for basis selection in relative-state interpretations; the consistently quantum origin ofthe Schmidt decomposition, which matches well the“pure quantum-mechanics” spirit of Everett’s proposalswhere the formalism of quantum mechanics supplies itsown interpretationd, has also been counted as an advan-tage sBarvinsky and Kamenshchik, 1995d. In an earlierwork, Deutsch s1985d attributed a fundamental role tothe Schmidt decomposition in relative-state interpreta-tions as defining an “interpretation basis” that imposesthe precise structure that is needed to give meaning toEverett’s basic concept.

However, as pointed out in Sec. III.E.4, emerging ba-sis states based on the instantaneous Schmidt states willfrequently have properties that are very different fromthose selected by the stability criterion and that are un-desirably nonclassical. For example, they may lack thespatial localization of the robustness-selected GaussianssStapp, 2002d. The question to what extent the Schmidtbasis states correspond to classical properties in Everett-

13For interesting quantitative results on the role of decoher-ence in brain processes, see Tegmark s2000d. Note, however,the sat least partiald rebuttal of Tegmark’s claims by Hagan etal. s2002d.



style interpretations was investigated in detail by Barv-insky and Kamenshchik s1995d. The authors study thesimilarity of the states selected by the Schmidt decom-position to coherent states si.e., minimum-uncertaintyGaussiansd that are chosen as the “yardstick states” rep-resenting classicality ssee also Eisert, 2004d. For the in-vestigated toy models it is found that only subsets of theEverett worlds corresponding to the Schmidt decompo-sition exhibit classicality in this sense; furthermore, thedegree of robustness of classicality in these branches isvery sensitive to the choice of the initial state and theinteraction Hamiltonian, such that classicality emergestypically only temporarily, and the Schmidt basis gener-ally lacks robustness under time evolution. Similar diffi-culties with the Schmidt-basis approach have been de-scribed by Kent and McElwaine s1997d.

These findings indicate that a selection criterion basedon robustness provides a much more meaningful, physi-cally transparent, and general rule for the selection ofquasiclassical branches in relative-state interpretations,especially with respect to its ability to lead to wave-function components representing quasiclassical proper-ties that can be reidentified over time swhich a simplediagonalization of the reduced density matrix at eachinstant of time does not, in general, allow ford.

2. Probabilities in Everett interpretations

Various attempts unrelated to decoherence have beenmade to find a consistent derivation of the Born prob-abilities sfor instance, Everett, 1957; Hartle, 1968; DeWitt, 1971; Graham, 1973; Geroch, 1984; Deutsch, 1999din the explicit or implicit context of a relative-state in-terpretation, but several arguments have been presentedthat show that these approaches fail.14 When the effectsof decoherence and environment-induced superselectionare included, it seems natural to identify the diagonalelements of the decohered reduced density matrix sinthe environment-superselected basisd with the set of pos-sible elementary events and to interpret the correspond-ing coefficients as relative frequencies of worlds sorminds, etc.d in the Everett theory, assuming a typicallyinfinite multitude of worlds, minds, etc. Since decoher-ence enables one to reidentify the individual localizedcomponents of the wave function over time sdescribing,for example, observers and their measurement outcomesattached to individual well-defined “worlds”d, this leadsto a natural interpretation of the Born probabilities asempirical frequencies.

However, decoherence cannot yield an actual deriva-tion of the Born rule sfor attempts in this direction, seeZurek, 1998; Deutsch, 1999d. As mentioned before, thisis so because the key elements of the decoherence pro-gram, the formalism and the interpretation of reduceddensity matrices and the trace rule, presume the Born

rule. Attempts to consistently derive probabilities fromreduced density matrices and from the trace rule aretherefore subject to the charge of circularity sZeh, 1997;Zurek, 2003ad. In Sec. III.F, we outlined a recent pro-posal by Zurek s2003bd that evades this circularity anddeduces the Born rule from envariance, a symmetryproperty of entangled systems, and from certain assump-tions about the connection between the state of the sys-tem S of interest, the state vector of the composite sys-tem SE that includes an environment E entangled with S,and probabilities of outcomes of measurements per-formed on S. Decoherence combined with this approachprovides a framework in which quantum probabilitiesand the Born rule can be given a rather natural motiva-tion, definition, and interpretation in the context ofrelative-state interpretations.

3. The “existential interpretation”

A well-known Everett-type interpretation that reliesheavily on decoherence has been proposed by Zureks1993, 1998; see also the recent reevaluation in Zurek,2004ad. This approach, termed the “existential interpre-tation,” defines the reality, or objective existence, of astate as the possibility of finding out what the state isand simultaneously leaving it unperturbed, similar to aclassical state.15 Zurek assigns a “relative objective exis-tence” to the robust states sidentified with elementary“events”d selected by the environmental stability crite-rion. By measuring properties of the system-environment interaction Hamiltonian and employing therobustness criterion, the observer can, at least in prin-ciple, determine the set of observables that can be mea-sured on the system without perturbing it and thus findout its “objective” state. Alternatively, the observer cantake advantage of the redundant records of the state ofthe system as monitored by the environment. By inter-cepting parts of this environment, for example, a frac-tion of the surrounding photons, he can determine thestate of the system essentially without perturbing it scf.also the related recent ideas of “quantum Darwinism”and the role of the environment as a “witness,” seeZurek, 2000, 2003a, 2004b; Ollivier et al., 2003d.16

Zurek emphasizes the importance of stable recordsfor observers, i.e., robust correlations between theenvironment-selected states and the memory states ofthe observer. Information must be represented physi-cally, and thus the “objective” state of the observer whohas detected one of the potential outcomes of a mea-surement must be physically distinct and objectively dif-ferent from the state of an observer who has recordedan alternative outcome ssince the record states can be

14See, for example, the critiques of Stein s1984d; Kent s1990d;Squires s1990d; Barnum et al. s2000d. However, also note thearguments of Wallace s2003bd and Gill s2003d, defending theapproach of Deutsch s1999d; see also Saunders s2002d.

15This intrinsically requires the notion of open systems, since,in isolated systems, the observer would need to know in ad-vance what observables commute with the state of the system,in order to perform a nondemolition measurement that avoidsrepreparing the state of the system.

16The partial ignorance is necessary to avoid redefinition ofthe state of the system.



determined from the outside without perturbing them—see the previous paragraphd. The different objectivestates of the observer are, via quantum correlations, at-tached to different branches defined by theenvironment-selected robust states; they thus ultimatelylabel the different branches of the universal state vector.This is claimed to lead to the perception of classicality;the impossibility of perceiving arbitrary superpositions isexplained via the quick suppression of interference be-tween different memory states induced by decoherence,where each sphysically distinctd memory state representsan individual observer identity.

A similar argument has been given by Zeh s1993d,who employs decoherence together with an simplicitdbranching process to explain the perception of definiteoutcomes:

fAgfter an observation one need not necessarilyconclude that only one component now exists butonly that only one component is observed. s…d Su-perposed world components describing the regis-tration of different macroscopic properties by the“same” observer are dynamically entirely indepen-dent of one another: they describe different ob-servers. s…d He who considers this conclusion ofan indeterminism or splitting of the observer’sidentity, derived from the Schrödinger equation inthe form of dynamically decoupling s“branching”dwave packets on a fundamental global configura-tion space, as unacceptable or “extravagant” mayinstead dynamically formalize the superfluous hy-pothesis of a disappearance of the “other” compo-nents by whatever method he prefers, but heshould be aware that he may thereby also createhis own problems: Any deviation from the globalSchrödinger equation must in principle lead to ob-servable effects, and it should be recalled that nonehave ever been discovered.

The existential interpretation has recently been con-nected to the theory of envariance ssee Zurek, 2004a,and Sec. III.Fd. In particular, the derivation of Born’srule based on envariance as outlined in Sec. III.F can berecast in the framework of the existential interpretationsuch that probabilities refer explicitly to the futurerecord state of an observer. The concept of such a prob-ability bears similarities to classical probability theorysfor more details on these ideas, see Zurek, 2004ad.

The existential interpretation continues Everett’s goalof interpreting quantum mechanics using the quantum-mechanical formalism itself. Zurek takes the standardno-collapse quantum theory “as is” and explores to whatextent the incorporation of environment-induced super-selection and decoherence sand recently also envari-anced could form a viable interpretation that would, witha minimal interpretive framework, be capable of ac-counting for the perception of definite outcomes and ofexplaining the origin and nature of probabilities.

D. Modal interpretations

The first type of modal interpretation was suggestedby van Fraassen s1973, 1991d, based on his program of“constructive empiricism,” which proposes to take onlyempirical adequacy, but not necessarily “truth,” as thegoal of science. Since then, a large number of interpre-tations of quantum mechanics have been suggested thatcan be considered as modal sfor a review and discussionof some of the basic properties and problems of suchinterpretations, see Clifton, 1996d.

In general, the approach of modal interpretations con-sists in weakening the orthodox e-e link by allowing forthe assignment of definite measurement outcomes evenif the system is not in an eigenstate of the observablerepresenting the measurement. In this way, one can pre-serve a purely unitary time evolution without the needfor an additional collapse postulate to account for defi-nite measurement results. Of course, this immediatelyraises the question of how physical properties that areperceived through measurements and measurement re-sults are connected to the state, since the bidirectionallink is broken between the eigenstate of the observableswhich corresponds to the physical propertyd and the ei-genvalue swhich represents the manifestation of thevalue of this physical property in a measurementd. Thegeneral goal of modal interpretations is then to specifyrules that determine a catalog of possible properties of asystem described by the density matrix r at time t. Twodifferent views are typically distinguished: a semanticapproach that only changes the way of talking about theconnection between properties and state; and a realisticview that provides a different specification of what thepossible properties of a system really are, given the statevector sor the density matrixd.

Such an attribution of possible properties must fulfillcertain requirements. For instance, probabilities for out-comes of measurements should be consistent with theusual Born probabilities of standard quantum mechan-ics; it should be possible to recover our experience ofclassicality in the perception of macroscopic objects; andan explicit time evolution of properties and their prob-abilities should be definable that is consistent with theresults of the Schrödinger equation. As we shall see inthe following, decoherence has frequently been em-ployed in modal interpretations to motivate and definerules for property ascription. Dieks s1994a, 1994bd hasargued that one of the central goals of modal ap-proaches is to provide an interpretation for decoher-ence.

1. Property assignment based on environment-inducedsuperselection

The intrinsic difficulty of modal interpretations is toavoid any ad hoc character of the property assignment,yet to find generally applicable rules that lead to a selec-tion of possible properties that include the determinateproperties of our experience. To solve this problem,various modal interpretations have embraced the resultsof the decoherence program. A natural approach would



be to employ the environment-induced superselection ofa preferred basis—since it is based on an entirely physi-cal and very general criterion snamely, the stability re-quirementd and has, for the cases studied, been shown togive results that agree well with our experience, thusmatching van Fraassen’s goal of empirical adequacy—toyield sets of possible quasiclassical properties associatedwith the correct probabilities.

Furthermore, since the decoherence program is basedsolely on Schrödinger dynamics, the task of defining atime evolution of the “property states” and their associ-ated probabilities that is in agreement with the results ofunitary quantum mechanics would presumably be easierthan in a model of property assignment in which the setof possibilities does not arise dynamically via theSchrödinger equation alone sfor a detailed proposal formodal dynamics of the latter type, see Bacciagaluppiand Dickson, 1999d. The need for explicit dynamics ofproperty states in modal interpretations is controversial.One can argue that it suffices to show that at each in-stant of time, the set of possibly possessed propertiesthat can be ascribed to the system is empirically ad-equate, in the sense of containing the properties of ourexperience, especially with respect to the properties ofmacroscopic objects sthis is essentially the view of, forexample, van Fraassen, 1973, 1991d. On the other hand,this cannot ensure that these properties behave overtime in agreement with our experience sfor instance,that macroscopic objects that are left undisturbed do notchange their position in space spontaneously in an ob-servable mannerd. In other words, the emergence ofclassicality is to be tied not only to determinate proper-ties at each instant of time, but also to the existence ofquasiclassical “trajectories” in property space. Since de-coherence allows one to reidentify components of thedecohered density matrix over time, this could be usedto derive property states with continuous, quasiclassicaltrajectorylike time evolution based on Schrödinger dy-namics alone. For some discussions of this approach, seeHemmo s1996d and Bacciagaluppi and Dickson s1999d.

The fact that the states emerging from decoherenceand the stability criterion are sometimes nonorthogonalor form a continuum will presumably be of even lessrelevance in modal interpretations than in Everett-styleinterpretations ssee Sec. IV.Cd, since the goal here issolely to specify sets of possible properties, of whichonly one set actually gets assigned to the system. Hencean overlap of the sets is not necessarily a problemsmodulo the potential difficulty of a straightforward as-signment of probabilities in such a situationd.

2. Property assignment based on instantaneous Schmidtdecompositions

Since it is usually rather difficult to determine explic-itly the robust “pointer states” through the stability sor asimilard criterion, it would not be easy to specify a gen-eral rule for property assignment based on environment-induced superselection. To simplify this situation, sev-eral modal interpretations have restricted themselves to

the orthogonal decomposition of the density matrix todefine the set of properties that can be assigned ssee, forinstance, Kochen, 1985; Dieks, 1989; Healey, 1989; Ver-maas and Dieks, 1995; Bub, 1997d. For example, the ap-proach of Dieks s1989d recognizes, by referring to thedecoherence program, the relevance of the environmentby considering a composite system-environment statevector and its diagonal Schmidt decomposition, ucl=ok

ÎpkufkSlufk

El, which always exists. Possible propertiesthat can be assigned to the system are then represented

by the Schmidt projectors Pk=lkufkSlkfk

Su. Although allterms are present in the Schmidt expansion sthat Diekscalls the “mathematical state”d, the “physical state” ispostulated to be given by only one of the terms, withprobability pk. A generalization of this approach to adecomposition into any number of subsystems has beendescribed by Vermaas and Dieks s1995d. In this sense,the Schmidt decomposition itself is taken to define aninterpretation of quantum mechanics. Dieks s1995d sug-gested a physical motivation for the Schmidt decompo-sition in modal interpretations based on the assumedrequirement of a one-to-one correspondence betweenthe properties of the system and its environment. For acomment on the violation of the property compositionprinciple in such interpretations, see the analysis of Clif-ton s1996d.

A central problem associated with the approach oforthogonal decomposition is that it is not at all clear thatthe properties determined by the Schmidt diagonaliza-tion represent the determinate properties of our experi-ence. As outlined in Sec. III.E.4, the states selected bythe sinstantaneousd orthogonal decomposition of the re-duced density matrix will in general differ from the ro-bust “pointer states” chosen by the stability criterion ofthe decoherence program and may have distinctly non-classical properties. That this will be the case especiallywhen the states selected by the orthogonal decomposi-tion are close to degeneracy has already been indicatedin Sec. III.E.4. It has also been explored in more detailin the context of modal interpretations by Bacciagaluppiet al. s1995d and Donald s1998d, who showed that in thecase of near degeneracy sas it typically occurs for mac-roscopic systems with many degrees of freedomd, the re-sulting projectors will be extremely sensitive to the pre-cise form of the state sBacciagaluppi et al., 1995d. Clearlysuch sensitivity is undesired since the projectors, andthus the properties of the system, will not be well be-haved under the inevitable approximations employed inphysics sDonald, 1998d.

3. Property assignment based on decompositions of thedecohered density matrix

Other authors therefore have appealed to the or-thogonal decomposition of the decohered reduced den-sity matrix sinstead of the decomposition of the instan-taneous density matrixd, which has led to noteworthyresults. When the system is represented by only a finite-dimensional Hilbert space, a discrete model of decoher-ence, the resulting states were indeed found to be typi-



cally close to the robust states selected by the stabilitycriterion sfor macroscopic systems, this typically meantlocalization in position spaced, unless again the finalcomposite state was very nearly degenerate sBacciaga-luppi and Hemmo, 1996; Bene, 2001; see also Sec.III.E.4d. Thus, in sufficiently nondegenerate cases, deco-herence can ensure that the definite properties selectedby modal interpretations of the Dieks-type will be ap-propriately close to the properties corresponding to theideal pointer states if the modal properties are based onthe orthogonal decomposition of the reduced decohereddensity matrix.

On the other hand, Bacciagaluppi s2000d showed thatin the more general and realistic case of an infinite-dimensional state space of the system, when one em-ploys a continuous model of decoherence snamely, thatof Joos and Zeh, 1985d, the predictions of the modalapproach sDieks, 1989; Vermaas and Dieks, 1995d andthose of decoherence can differ significantly. It was dem-onstrated that the definite properties obtained from theorthogonal decomposition of the decohered density ma-trix were highly delocalized sthat is, smeared out overthe entire spread of the stated, although the coherencelength of the density matrix itself was shown to be verysmall, so that decoherence indicated very localized prop-erties. Thus, based on these results sand similar ones ofDonald, 1998d, decoherence can be used to argue for thephysical inadequacy of the rule for the assignment ofdefinite properties as proposed by Dieks s1989d and Ver-maas and Dieks s1995d.

More generally, if the definite properties selected bythe modal interpretation fail to mesh with the results ofdecoherence sin particular, when they also lack the de-sired classicality and correspondence to the determinateproperties of our experienced, we are given reason todoubt whether the proposed rules for property assign-ment have sufficient physical motivation, legitimacy, orgenerality.

4. Concluding remarks

There are many different proposals that can begrouped under the heading of modal interpretations.They all share the problem of motivating and verifying aconsistent system of property assignment. Using the ro-bust pointer states selected by interaction with the envi-ronment and by the stability criterion is a step in theright direction, but the difficulty remains to derive a gen-eral rule for property assignment from this method thatwould yield explicitly the sets of possibilities in everysituation. In certain cases, for example, close to degen-eracy and in Hilbert spaces of infinite dimension, thesimpler approach of deriving the possible propertiesfrom the orthogonal decomposition of the decohered re-duced density matrix fails to yield the sharply localized,quasiclassical pointer states as selected by environmen-tal robustness criteria. These are the cases in which de-coherence can play a vital role in helping to identifyinadequate rules for property assignment in modal inter-pretations.

E. Physical collapse theories

The basic idea of physical collapse theories is to intro-duce an explicit modification of the Schrödinger timeevolution to achieve a physical mechanism for state-vector reduction sfor an extensive recent review, seeBassi and Ghirardi, 2003d. This is in general motivatedby a “realist” interpretation of the state vector, that is,the state vector is directly identified with a physicalstate, which then requires reduction to one of the termsin the superposition to establish equivalence to the ob-served determinate properties of physical states, at leastas far as the macroscopic realm is concerned.

The first proposals for theories of this type were madeby Pearle s1976, 1979, 1982d and Gisin s1984d, who de-veloped dynamical reduction models that modify unitarydynamics such that a superposition of quantum statesevolves continuously into one of its terms ssee also thereview by Pearle, 1999d. Typically, terms representingexternal white noise are added to the Schrödinger equa-tion, causing the squared amplitudes ucnstdu2 in the state-vector expansion uCstdl=oncnstducnl to fluctuate ran-domly in time, while maintaining the normalizationcondition onucnstdu2=1 for all t. This process is known asstochastic dynamical reduction. Eventually one ampli-tude ucnstdu2→1, while all other squared coefficients →0sthe “gambler’s ruin game” mechanismd, where ucnstdu2→1 with probability ucnst=0du2 sthe squared coefficientsin the initial precollapse state-vector expansiond inagreement with the Born probability interpretation ofthe expansion coefficients.

These early models exhibit two main difficulties. First,the preferred-basis problem: What determines the termsin the state-vector expansion into which the state vectorgets reduced? Why does reduction lead to precisely thedistinct macroscopic states of our experience and notsuperpositions thereof? Second, how can one accountfor the fact that the effectiveness of collapsing superpo-sitions increases when going from microscopic to macro-scopic scales?

These problems motivated spontaneous localizationmodels, initially proposed by Ghirardi, Rimini, and We-ber sGRW; Ghirardi et al., 1986d. Here state-vector re-duction is not implemented as a dynamical process si.e.,as a continuous evolution over timed, but instead occursinstantaneously and spontaneously, leading to a spatiallocalization of the wave function. To be precise, theN-particle wave function csx1 ,… ,xNd is at random inter-vals multiplied by a Gaussian of the form expf−sX−xkd2 /2D2g sthis process is often called a “hit” or a“jump”d, and the resulting product is subsequently nor-malized. The occurrence of these hits is not explained,but rather postulated as a new fundamental physicalmechanism. Both the coordinate xk and the “center ofthe hit” X are chosen at random, but the probability fora specific X is postulated to be given by the squaredinner product of csx1 ,… ,xNd with the Gaussian sandtherefore hits are more likely to occur where ucu2,viewed as a function of xk only, is larged. The mean fre-quency n of hits for a single microscopic particle is cho-



sen so as to effectively preserve unitary time evolutionfor microscopic systems, while ensuring that for macro-scopic objects containing a very large number N of par-ticles the localization occurs rapidly son the order ofNnd, in such a way as to preclude the persistence of spa-tially separated macroscopic superpositions ssuch as thepointer’s being in a superpositon of “up” and “down”don time scales shorter than realistic observations couldresolve. Ghirardi et al. s1986d choose n<10−16 s−1, so amacrosopic system with N<1023 particles undergoes lo-calization on average every 10−7 s. Inevitable coupling tothe environment can in general be expected to lead to afurther drastic increase of N and therefore to an evenhigher localization rate. Note, however, that the localiza-tion process itself is independent of any interaction withthe environment, in sharp contrast to the decoherenceapproach.

Subsequently, the ideas of stochastic dynamical reduc-tion and GRW theory were combined into continuousspontaneous localization models sPearle, 1989; Ghirardiet al., 1990d in which localization of the GRW-type canbe shown to emerge from a nonunitary, nonlinear Itôstochastic differential equation, namely, the Schrödingerequation augmented by spatially correlated Brownianmotion terms ssee also Diósi, 1988, 1989d. The particularchoice of stochastic term determines the preferred basis.Frequently, the stochastic term has been based on themass density, which yields a GRW-type spatial localiza-tion sDiósi, 1989; Pearle, 1989; Ghirardi et al., 1990d, butstochastic terms driven by the Hamiltonian, leading to areduction on an energy basis, have also been studiedsBedford and Wang, 1975, 1977; Milburn, 1991; Percival,1995, 1998; Hughston, 1996; Fivel, 1997; Adler and Hor-witz, 2000; Adler et al., 2001; Adler, 2002d. If we focus onthe first type of term, the Ghirardi-Rimini-Weber theoryand continuous spontaneous localization become phe-nomenologically similar, and we shall refer to themjointly as “spontaneous localization” models in the fol-lowing discussion whenever it is unnecessary to distin-guish them explicitly.

1. The preferred-basis problem

Physical reduction theories typically remove wave-function collapse from the restrictive context of the or-thodox interpretation swhere the external observer arbi-trarily selects the measured observable and thusdetermines the preferred basisd, and rather understandreduction as a universal mechanism that acts constantlyon every state vector regardless of an explicit measure-ment situation. In view of this, it is particularly impor-tant to provide a definition for the states into which thewave function collapses.

As mentioned before, the original stochastic dynami-cal reduction models suffer from this preferred-basisproblem. Taking into account environment-induced su-perselection of a preferred basis could help resolve thisissue. Decoherence has been shown to occur, especiallyfor mesoscopic and macroscopic objects, on extremelyshort time scales, and thus would presumably be able to

bring about basis selection much faster than the timerequired for dynamical fluctuations to establish a “win-ning” expansion coefficient.

In contrast, the GRW theory solves the preferred-basis problem by postulating a mechanism that leads toreduction to a particular state vector in an expansion ona position basis, i.e., position is assumed to be the uni-versal preferred basis. State-vector reduction thenamounts to simply modifying the functional shape of theprojection of the state vector ucl onto the position basiskx1 ,… ,xNu. This choice can be motivated by the insightthat essentially all shumand observations must begrounded in a position measurement.17

On the one hand, the selection of position as the pre-ferred basis is supported by the decoherence program,since physical interactions frequently are governed bydistance-dependent laws. Given the stability criterion ora similar requirement, this leads to position as the pre-ferred observable. In this sense, decoherence provides aphysical motivation for the assumption of the GRWmodel. On the other hand, it makes this assumption ap-pear as too restrictive since it cannot account for cases inwhich position is not the preferred basis—for instance,in microscopic systems where typically energy is the ro-bust observable, or in the superposition of smacroscopicdcurrents in SQUID’s. The GRW model simply excludessuch cases by choosing the parameters of the spontane-ous localization process such that microscopic systemsremain generally unaffected by any state-vector reduc-tion. The basis selection approach proposed by the de-coherence program is therefore much more general andalso avoids the ad hoc character of the GRW theory byallowing for a range of preferred observables and moti-vating their choice on physical grounds.

A similar argument can be made with respect to thecontinuous spontaneous localization approach. Here,one essentially preselects a preferred basis through theparticular choice of the stochastic terms added to theSchrödinger equation. This allows for a greater range ofpossible preferred bases, for instance, by combiningterms driven by the Hamiltonian and by the mass den-sity, leading to a competition between localization in en-ergy and position space scorresponding to the two mostfrequently observed eigenstatesd. Nonetheless, any par-ticular choice of terms will again be subject to the chargeof possessing an ad hoc flavor, in contrast to the physicaldefinition of the preferred basis derived from the struc-ture of the unmodified Hamiltonian as suggested byenvironment-induced selection.

2. Simultaneous presence of decoherence andspontaneous localization

Since decoherence can be considered as an omnipres-ent phenomenon that has been extensively verified both

17This measurement may ultimately occur only in the brain ofthe observer; see the objection to the GRW model by Albertand Vaidman s1989d. With respect to the general preferencefor position as the basis of measurements, see also the com-ment by Bell s1982d.



theoretically and experimentally, the assumption that aphysical collapse theory holds means that the evolutionof a system must be guided by both decoherence effectsand the reduction mechanism.

Let us first consider the situation in which decoher-ence and the localization mechanism act constructivelyin the same direction, i.e., towards a common preferredbasis. This raises the question in which order these twoeffects influence the evolution of the system sBacciaga-luppi, 2003ad. If localization occurs on a shorter timescale than environment-induced superselection of a pre-ferred basis and suppression of local interference, deco-herence will in most cases have very little influence onthe evolution of the system, since typically the systemwill already have evolved into a reduced state. Con-versely, if decoherence effects act more quickly on thesystem than the localization mechanism, the interactionwith the environment will presumably lead to the prepa-ration of quasiclassical robust states that are subse-quently chosen by the localization mechanism. Aspointed out in Sec. III.D, decoherence usually occurs onextremely short time scales, which can be shown to besignificantly smaller than the action of the spontaneouslocalization process for most cases sfor studies related tothe GRW model, see Tegmark, 1993 and Benatti et al.,1995d. This indicates that decoherence will typically playan important role even in the presence of physical wave-function reduction.

The second case occurs when decoherence leads tothe selection of a different preferred basis than the re-duction basis specified by the localization mechanism.As remarked by Bacciagaluppi s2003a, 2003bd in thecontext of the GRW theory, one might then imagine thecollapse either to occur only at the level of the environ-ment swhich would then serve as an amplifying and re-cording device with different localization propertiesthan the system under studyd, or to lead to an explicitcompetition between decoherence and localization ef-fects.

3. The tails problem

The clear advantage of physical collapse models overthe consideration of decoherence-induced effects alonefor a solution to the measurement problem lies in thefact that an actual state reduction is achieved such thatone may be tempted to conclude that at the conclusionof the reduction process the system actually is in a de-terminate state. However, all collapse models achieveonly an approximate s“for all practical purposes”d reduc-tion of the wave function. In the case of dynamical re-duction models, the state will always retain small inter-ference terms for finite times. Similarly, in the GRWtheory the width D of the multiplying Gaussian cannotbe made arbitrarily small, and therefore the reducedwave packet cannot be made infinitely sharply localizedin position space, since this would entail an infinitelylarge energy gain by the system via the time-energy un-certainty relation, which would certainly show up ex-perimentally sGhirardi et al., 1986, chose D<10−5 cmd.

This need for only an approximate reduction leads towave function “tails” sAlbert and Loewer, 1996d, that is,in any region in space and at any time t.0, the wavefunction will remain nonzero if it has been nonzero att=0 sbefore the collapsed, and thus there will be always apart of the system that is not “here.”

Physical collapse models that achieve reduction only“for all practical purposes” require a modification,namely, a weakening, of the orthodox e-e link to allowone to speak of the system’s actually being in a definitestate, and thereby to ensure the objective attribution ofdeterminate properties to the system.18 In this sense, col-lapse models are as much “just fine for all practical pur-poses” sto paraphrase Bell, 1990d as decoherence is,where perfect orthogonality of the environment states isonly attained as t→`. The severity of the consequences,however, is not equivalent for the two strategies. Sincecollapse models directly change the state vector, a singleoutcome is at least approximately selected, and it onlyrequires a “slight” weakening of the e-e link to makethis state of affairs correspond to the sobjectived exis-tence of a determinate physical property. In the case ofdecoherence, the lack of a precise destruction of inter-ference terms is not the main problem; even if exactorthogonality of the environment states were ensured atall times, the resulting reduced density matrix would stillrepresent an improper mixture, with no outcome havingbeen singled out according to the e-e link. This would bethe case regardless of whether the e-e link were ex-pressed in the strong or weakened form, and we wouldstill have to supply some additional interpretative frame-work to explain our perception of outcomes ssee also thecomment by Ghirardi et al., 1987d.

4. Connecting decoherence and collapse models

It was realized early that there exists a striking formalsimilarity of the equations that govern the time evolu-tion of density matrices in the GRW approach and inmodels of decoherence. For example, the GRW equa-tion for a single free mass point reads fGhirardi et al.,1986, Eq. s3.5dg

i] rsx,x8,td

] t=

1

2mF ]2

] x2 −]2

] x82Gr − iLsx − x8d2r , s4.1d

where the second term on the right-hand side accountsfor the destruction of spatially separated interferenceterms. A simple model for environment-induced deco-herence yields a very similar equation fJoos and Zeh,1985, Eq. s3.75d; see also the comment by Joos, 1987g.Thus the physical justification for an ad hoc postulate ofan explicit reduction-inducing mechanism could bequestioned sof course modulo the important interpretivedifference between the approximately proper ensemblesarising from collapse models and the improper en-

18It should be noted, however, that such “fuzzy” e-e links mayin turn lead to difficulties, as the discussion of Lewis’s “count-ing anomaly” has shown sLewis, 1997d.



sembles resulting from decoherence; see also Ghirardi etal., 1987d. More constructively, the similarity of the gov-erning equations might enable one to choose the freeparameters in collapse models on physical groundsrather than on the basis of empirical adequacy. Con-versely, this similarity can also be viewed as leading to a“protection” of physical collapse theories from empiricaldisproof. This is so because the inevitable and ubiqui-tous interaction with the environment will always, for allpractical purposes of observation sthat is, of statisticalpredictiond, result in slocald density matrices that are for-mally very similar to those of collapse models. What ismeasured is not the state vector itself, but the probabil-ity distribution of outcomes, i.e., values of a physicalquantity and their frequency, and this information isequivalently contained in the state vector and the den-sity matrix. Measurements with their intrinsically localcharacter will presumably be unable to distinguish be-tween the probability distribution given by the deco-hered reduced density matrix and the probability distri-bution defined by an sapproximatelyd proper mixtureobtained from a physical collapse. In other words, aslong as the free parameters of collapse theories are cho-sen in agreement with those determined from decoher-ence, models for state-vector reduction can be expectedto be empirically adequate since decoherence is an ef-fect that will be present with near certainty in every re-alistic sespecially macroscopicd physical system.

One might of course speculate that the simultaneouspresence of both decoherence and reduction effectsmight actually allow for an experimental disproof of col-lapse theories by preparing states that differ in an ob-servable manner from the predictions of the reductionmodels.19 If we acknowledge the existence of interpreta-tions of quantum mechanics that employ onlydecoherence-induced suppression of interference to ex-plain the perception of apparent collapses sas is, for ex-ample, claimed by the “existential interpretation” ofZurek, 1993, 1998; see Sec. IV.C.3d, we will not be ableto distinguish experimentally between a “true” collapseand a mere suppression of interference as explained bydecoherence. Instead, an experimental situation is re-quired in which the collapse model predicts a collapse,but in which no suppression of interference through de-coherence arises. Again, the problem in the realizationof such an experiment is that it is very difficult to shielda system from decoherence effects, especially since wewill typically require a mesoscopic or macroscopic sys-tem in which the reduction is efficient enough to be ob-served. For example, based on explicit numerical esti-mates, Tegmark s1993d has shown that decoherence dueto scattering of environmental particles such as air mol-ecules or photons will have a much stronger influence

than the proposed GRW effect of spontaneous localiza-tion ssee also Benatti et al., 1995; Bassi and Ghirardi,2003; for different results for energy-driven reductionmodels, cf. Adler, 2002d.

5. Summary and outlook

Decoherence has the distinct advantage of being de-rived directly from the laws of standard quantum me-chanics, whereas current collapse models are required topostulate their reduction mechanism as a new funda-mental law of nature. On the other hand, collapse mod-els yield, at least for all practical purposes, proper mix-tures, so they are capable of providing an “objective”solution to the measurement problem. The formal simi-larity between the time evolution equations of the col-lapse and decoherence models nourishes hopes that thepostulated reduction mechanisms of collapse modelscould possibly be derived from the ubiquituous and in-evitable interaction of every physical system with its en-vironment and the resulting decoherence effects. Wemay therefore regard collapse models and decoherencenot as mutually exclusive alternatives for a solution tothe measurement problem, but rather as potential can-didates for a fruitful unification. For a vague proposalalong these lines, see Pessoa s1998d; cf. also Diósi s1989dand Pearle s1999d for speculations that quantum gravitymight act as a collapse-inducing universal “environ-ment.”

F. Bohmian mechanics

Bohm’s approach sBohm, 1952; Bohm and Bub, 1966;Bohm and Hiley, 1993d is a modification of de Broglie’ss1930d original “pilot-wave” proposal. In Bohmian me-chanics, a system containing N snonrelativisticd particlesis described by a wave function cstd and the configura-tion Qstd= „q1std ,… ,qNstd…PR3N of particle positionsqistd, i.e., the state of the system is represented by sc ,Qdfor each instant t. The evolution of the system is guidedby two equations. The wave function cstd is transformedas usual via the standard Schrödinger equation,

i"s] /]tdc=Hc, while the particle positions qistd of theconfiguration Qstd evolve according to the “guidingequation”

dqi

dt= vi

csq1,…,qNd ;"

miIm

c*¹qic

c*csq1,…,qNd , s4.2d

where mi is the mass of the ith particle. Thus the par-ticles follow determinate trajectories described by Qstd,with the distribution of Qstd being given by the quantumequilibrium distribution r= ucu2.

1. Particles as fundamental entities

Bohm’s theory has been criticized for ascribing funda-mental ontological status to particles. General argu-ments against particles on a fundamental level of anyrelativistic quantum theory have been frequently givenssee, for instance, Malament, 1996, and Halvorson and

19For proposed experiments to detect the GRW collapse, see,for example, Squires s1991d and Rae s1990d. For experimentsthat could potentially demonstrate deviations from the predic-tions of quantum theory when dynamical state-vector reduc-tion is present, see Pearle s1984, 1986d.



Clifton, 2002d.20 Moreover, and this is the point wewould like to discuss in this section, it has been arguedthat the appearance of particles s“discontinuities inspace”d could be derived from the continuous process ofdecoherence, leading to claims that no fundamental roleneed be attributed to particles sZeh, 1993, 1999, 2003d.Based on decohered density matrices of mesoscopic andmacroscopic systems that essentially always representquasiensembles of narrow wave packets in positionspace, Zeh s1993, p. 190d holds that such wave packetscan be viewed as representing individual “particle”positions:21

All particle aspects observed in measurements ofquantum fields slike spots on a plate, tracks in abubble chamber, or clicks of a counterd can be un-derstood by taking into account this decoherenceof the relevant local si.e., subsystemd density ma-trix.

The first question is then whether a narrow wave packetin position space can be identified with the subjectiveexperience of a “particle.” The answer appears to beyes: our notion of “particles” hinges on the property oflocalizability, i.e., the definition of a region of space VPR3 in which the system sthat is, the support of thewave functiond is entirely contained. Although the na-ture of the Schrödinger dynamics implies that any wavefunction will have nonvanishing support s“tails”d outsideof any finite spatial region V and therefore exact local-izatibility will never be achieved, we only need to de-mand approximate localizability to account for our ex-perience of particle aspects.

However, to interpret the ensembles of narrow wavepackets resulting from decoherence as leading to theperception of individual particles, we must embed stan-dard quantum mechanics swith decoherenced into an ad-ditional interpretive framework that explains why onlyone of the wave packets is perceived;22 that is, we doneed to add some interpretive rule to get from the im-proper ensemble emerging from decoherence to the per-ception of individual terms, so decoherence alone doesnot necessarily make Bohm’s particle concept superflu-ous. But it suggests that the postulate of particles as fun-damental entities could be unnecessary, and taken to-gether with the difficulties in reconciling such a particle

theory with a relativistic quantum field theory, Bohm’s apriori assumption of particles at a fundamental level ofthe theory appears seriously challenged.

2. Bohmian trajectories and decoherence

A well-known property of Bohmian mechanics is thefact that its trajectories are often highly nonclassicalssee, for example, Bohm and Hiley, 1993; Holland, 1993;Appleby, 1999ad. This poses the serious problem of howBohm’s theory can explain the existence of quasiclassicaltrajectories on a macroscopic level.

Bohm and Hiley s1993d considered the scattering of abeam of environmental particles on a macroscopic sys-tem, a process that is known to give rise to decoherencesJoos and Zeh, 1985; Joos et al., 2003d. The authors dem-onstrate that this scattering yields quasiclassical trajecto-ries for the system. It has further been shown that, forisolated systems, the Bohm theory will typically not givethe correct classical limit sAppleby, 1999ad. It was thussuggested that the inclusion of the environment and ofthe resulting decoherence effects might be helpful in re-covering quasiclassical trajectories in Bohmian mechan-ics sAppleby, 1999b; Zeh, 1999; Allori, 2001; Allori andZanghì, 2001; Allori et al., 2002; Sanz and Borondo,2003d.

We mentioned before that the interaction between amacroscopic system and its environment will typicallylead to a rapid approximate diagonalization of the re-duced density matrix in position space, and thus to spa-tially localized wave packets that follow sapproximatelydHamiltonian trajectories. fThis observation also pro-vides a physical motivation for the choice of position asthe fundamental preferred basis in Bohm’s theory, inagreement with Bell’s s1982d well-known comment that“in physics the only observations we must consider areposition observations, if only the positions of instrumentpointers.”g The intuitive step is then to associate thesetrajectories with the particle trajectories Qstd of theBohm theory. As pointed out by Bacciagaluppi s2003bd,a great advantage of this strategy lies in the fact that thesame approach would allow for a recovery of both quan-tum and classical phenomena.

However, a careful analysis by Appleby s1999bdshowed that this decoherence-induced diagonalization inthe position basis alone will in general not suffice toyield quasiclassical trajectories in Bohm’s theory; onlyunder certain additional assumptions will processes thatlead to decoherence also give correct quasiclassicalBohmian trajectories for macroscopic systems sApplebydescribed the example of the long-time limit of a systemthat has initially been prepared in an energy eigenstated.Interesting results were also reported by Allori and co-workers sAllori, 2001; Allori and Zanghì, 2001; Allori etal., 2002d. They demonstrated that decoherence effectscan play the role of preserving classical properties ofBohmian trajectories. Furthermore, they showed thatwhile in standard quantum mechanics it is important tomaintain narrow wave packets to account for the emer-gence of classicality, the Bohmian description of a sys-

20On the other hand, there are proposals for a “Bohmianmechanics of quantum fields,” i.e., a theory that embeds quan-tum field theory into a Bohmian-style framework sDürr et al.,2003, 2004d.

21Schrödinger s1926d had made an attempt into a similar di-rection but had failed since the Schrödinger equation tends tocontinuously spread out any localized wave packet when it isconsidered as describing an isolated system. The inclusion ofan interacting environment and thus decoherence counteractsthe spread and opens up the possibility of maintaining narrowwave packets over time sJoos and Zeh, 1985d.

22Zeh himself, like Zurek s1998d, adheres to an Everett-stylebranching to which distinct observers are attached sZeh, 1993d,see also the quote in Sec. IV.C.



tem by both its wave function and its configuration al-lows for the derivation of quasiclassical behavior fromhighly delocalized wave functions. Sanz and Borondos2003d studied the double-slit experiment in the frame-work of Bohmian mechanics and in the presence of de-coherence and showed that even when coherence is fullylost, and thus interference is absent, nonlocal quantumcorrelations remain that influence the dynamics of theparticles in the Bohm theory, demonstrating that in thisexample decoherence does not suffice to achieve theclassical limit in Bohmian mechanics.

In conclusion, while the basic idea of employingdecoherence-related processes to yield the correct clas-sical limit of Bohmian trajectories seems reasonable,many details of this approach still need to be workedout.

G. Consistent-histories interpretations

The consistent- sor decoherent-d histories approachwas introduced by Griffiths s1984, 1993, 1996d and fur-ther developed by Omnès s1988a, 1988b, 1988c, 1990,1992, 1994, 2002d, Gell-Mann and Hartle s1990, 1991a,1991b, 1993d, Dowker and Halliwell s1992d, and others.Reviews of the program can be found in the papers byOmnès s1992d and Halliwell s1993, 1996d; thoughtful cri-tiques investigating key features and assumptions of theapproach have been given, for example, by d’Espagnats1989d, Dowker and Kent s1995, 1996d, Kent s1998d, andBassi and Ghirardi s1999d. The basic idea of theconsistent-histories approach is to eliminate the funda-mental role of measurements in quantum mechanics,and instead study quantum histories, defined as se-quences of events represented by sets of time-orderedprojection operators, and to assign probabilities to suchhistories. The approach was originally motivated byquantum cosmology, i.e., the study of the evolution ofthe entire universe, which, by definition, represents aclosed system. Therefore no external observer swhich is,for example, an indispensable element of the Copen-hagen interpretationd can be invoked.

1. Definition of histories

We assume that a physical system S is described by adensity matrix r0 at some initial time t0 and define asequence of arbitrary times t1, t2, ¯ , tn with t1. t0.For each time point ti in this sequence, we consider an

exhaustive set Psid= hPai

sidstid uai=1¯mij, 1ø iøn, of mu-

tually orthogonal Hermitian projection operators Pai

sidstid,obeying

oai

Pai

sidstid = 1, Pai

sidstidPbi

sidstid = daibiPai

sidstid , s4.3d

and evolving, using the Heisenberg picture, according to

Pai

sidstd = U†st0,tdPai

sidst0dUst0,td , s4.4d

where Ust0 , td is the operator that dynamically propa-gates the state vector from t0 to t.

A possible, “maximally fine-grained” history is de-fined by the sequence of times t1, t2, ¯ , tn and by thechoice of one projection operator in the set Psid for eachtime point ti in the sequence, i.e., by the set

Hhaj = hPa1

s1dst1d,Pa2

s2dst2d,…,Pan

sndstndj . s4.5d

We also define the set H= hHhajj of all possible historiesfor a given time sequence t1, t2, ¯ , tn. The naturalinterpretation of a history Hhaj is then to take it as aseries of propositions of the form “the system S was, at

time ti, in a state of the subspace spanned by Pai

sidstid.”Maximally fine-grained histories can be combined to

form “coarse-grained” sets which assign to each timepoint ti a linear combination

Qbi

sidstid = oai

pai

sidPai

sidstid, pai

sid P h0,1j s4.6d

of the original projection operators Pai

sidstid.

So far, the projection operators Pai

sidstid chosen at a cer-tain instant ti in time in order to form a history Hhaj wereindependent of the choice of the projection operators atearlier times t0, t, ti in Hhaj. This situation was gener-alized by Omnès s1988a, 1988b, 1988c, 1990, 1992d toinclude “branch-dependent” histories of the form sseealso Gell-Mann and Hartle, 1993d

Hhaj = hPa1

s1dst1d,Pa2

s2,a1dst2d,…,Pan

sn,a1,…,an−1dstndj . s4.7d

2. Probabilities and consistency

In standard quantum mechanics, we can always assignprobabilities to single events, represented by the eigen-

states of some projection operator Psidstd, via the rule

psi,td = TrfPsid†stdrst0dPsidstdg . s4.8d

The natural extension of this formula to the calculationof the probability psHhajd of a history Hhaj is given by

psHhajd = Dsa,ad , s4.9d

where the so-called decoherence functional Dsa ,bd is de-fined by sGell-Mann and Hartle, 1990d

Dsa,bd = TrfPan

sndstnd ¯ Pa1

s1dst1dr0Pb1

s1dst1d ¯ Pbn

sndstndg .

s4.10d

If we instead work in the Schrödinger picture, the deco-herence functional is



Dsa,bd = TrfPan

sndUstn−1,tnd ¯ Pa1

s1drst1dPb1

s1d¯ U†stn−1,tndPbn

sndstndg . s4.11d

Consider now the coarse-grained history that arises from a combination of the two maximally fine-grained historiesHhaj and Hhbj,

Hha∨bj = hPa1

s1dst1d + Pb1

s1dst1d,Pa2

s2dst2d + Pb2

s2dst2d,…,Pan

sndstnd + Pbn

sndstndj . s4.12d

We interpret each combined projection operator Pai

sidstid+ Pbi

sidstid as stating that, at time ti, the system was in the range

described by the union of Pai

sidstid and Pbi

sidstid. Accordingly, we would like to require that the probability for a historycontaining such a combined projection operator be equivalently calculable from the sum of the probabilities of the

two histories containing the individual projectors Pai

sidstid and Pbi

sidstid, that is,

TrfPan

sndstnd ¯ „Pai

sidstid + Pbi

sidstid… ¯ Pa1

s1dst1dr0Pa1

s1dst1d ¯ „Pai

sidstid + Pbi

sidstid… ¯ Pan

sndstndg

=!

TrfPan

sndstnd ¯ Pai

sidstid ¯ Pa1

s1dst1dr0Pa1

s1dst1d ¯ Pai

sidstid ¯ Pan

sndstndg

+ TrfPan

sndstnd ¯ Pbi

sidstid ¯ Pa1

s1dst1dr0Pa1

s1dst1d ¯ Pbi

sidstid ¯ Pan

sndstndg .

It can be easily shown that this relation holds if and only if

RehTrfPan

sndstnd ¯ Pai

sidstid ¯ Pa1

s1dst1dr0Pa1

s1dst1d ¯ Pbi

sidstid ¯ Pan

sndstndgj = 0 if ai Þ bi. s4.13d

Generalizing this two-projector case to the coarse-grained history Hha∨bj of Eq. s4.12d, we arrive at thessufficient and necessaryd consistency condition for twohistories Hhaj and Hhbj sGriffiths, 1984; Omnès, 1990,1992d,

RefDsa,bdg = da,bDsa,ad . s4.14d

If this relation is violated, the usual sum rule for calcu-lating probabilities does not apply. This situation ariseswhen quantum interference between the two combinedhistories Hhaj and Hhbj is present. Therefore, to ensurethat the standard laws of probability theory also hold forcoarse-grained histories, the set H of possible historiesmust be consistent in the above sense.

However, Gell-Mann and Hartle s1990d have pointedout that when decoherence effects are included to modelthe emergence of classicality, it is more natural to re-quire

Dsa,bd = da,bDsa,ad . s4.15d

Condition s4.14d has often been referred to as weak de-coherence, and Eq. s4.15d as medium decoherence sfor aproposal of a criterion for strong decoherence, see Gell-Mann and Hartle, 1998d. The set H of histories is calledconsistent sor decoherentd when all its members Hhaj ful-fill the consistency condition, Eqs. s4.14d or s4.15d, i.e.,when they can be regarded as independent snoninterfer-ingd.

3. Selection of histories and classicality

Even when the stronger consistency criterion s4.15d isimposed on the set H of possible histories, the number of

mutually incompatible consistent histories remains rela-tively large sd’Espagnat, 1989; Dowker and Kent, 1996d.It is not at all clear a priori that at least some of thesehistories necessarily represent any meaningful set ofpropositions about the world of our observation. Even ifa collection of such “meaningful” histories is found, itleaves open the question how to select such histories andwhich additional criteria would need to be invoked.

Griffiths’s s1984d original aim in formulating the con-sistency criterion was only to allow for a consistent de-scription of sequences of events in closed quantum sys-tems without running into logical contradictions.23

Commonly, however, consistency has also been tied tothe emergence of classicality. For example, the consis-tency criterion corresponds to the demand for the ab-sence of quantum interference—a property ofclassicality—between two combined histories. On theother hand, it has become clear that most consistent his-tories are in fact flagrantly nonclassical sGell-Mann andHartle, 1990, 1991b; Albrecht, 1993; Paz and Zurek,1993; Zurek, 1993; Dowker and Kent, 1995, 1996d. For

instance, when the projection operators Pai

sidstid are cho-sen to be the time-evolved eigenstates of the initial den-sity matrix rst0d, the consistency condition will automati-cally be fulfilled, yet the histories composed of theseprojection operators have been shown to result in highly

23However, Goldstein s1998d used a simple example to arguethat the consistent-histories approach can lead to contradic-tions with respect to a combination of joint probabilities, evenif the consistency criterion is imposed; see also the subsequentexchange of letters in the February 1999 issue of Physics To-day.



nonclassical macroscopic superpositions when applied tostandard examples such as quantum measurement orSchrödinger’s cat. This demonstrates that the consis-tency condition cannot serve as a sufficient criterion forclassicality.

4. Consistent histories of open systems

Various authors have appealed to interaction with theenvironment and the resulting decoherence effects indefining additional criteria that would select quasiclassi-cal histories and would also lead to a physical motivationfor the consistency criterion ssee, for example, Gell-Mann and Hartle, 1990, 1998; Albrecht, 1992, 1993;Dowker and Halliwell, 1992; Finkelstein, 1993; Paz andZurek, 1993; Twamley, 1993b; Zurek, 1993; Anastopou-los, 1996; Halliwell, 2001d. This approach intrinsically re-quires the notion of local, open systems and the split ofthe universe into subsystems, in contrast to the originalaim of the consistent-histories approach to describe theevolution of a single closed, undivided system stypicallythe entire universed. The decoherence-based studiesthen assume the usual decomposition of the total Hil-bert space H into a space HS, corresponding to the sys-tem S, and HE of an environment E. One then describesthe histories of the system S by employing projectionoperators that act only on the system, i.e., that are of the

form Pai

sidstid ^ IE, where Pai

sidstid acts only on HS and IE isthe identity operator in HE.

This raises the question of when, i.e., under which cir-cumstances, the reduced density matrix rS=TrErSE of thesystem S suffices to calculate the decoherence func-tional. The reduced density matrix arises from a nonuni-tary trace over E at every time point ti, whereas the de-coherence functional of Eq. s4.11d employs the full,unitarily evolving density matrix rSE for all times ti, tfand only applies a nonunitary trace operation sover bothS and Ed at the final time tf. Paz and Zurek s1993d haveanswered this srather technicald question by showingthat the decoherence functional can be expressed en-tirely in terms of the reduced density matrix if the timeevolution of the reduced density matrix is independentof the correlations dynamically formed between the sys-tem and the environment. A necessary sbut not alwayssufficientd condition for this requirement to be satisfiedis given by demanding that the reduced dynamics begoverned by a master equation that is local in time.

When a “reduced” decoherence functional exists, atleast to a good approximation, i.e., when the reduceddynamics are sufficiently insensitive to the formation ofsystem-environment correlations, the consistency ofwhole-universe histories, described by a unitarily evolv-ing density matrix rSE and sequences of projection op-

erators of the form Pai

sidstid ^ IE, will be directly related tothat of open-system histories, represented by a nonuni-tarily evolving reduced density matrix rSstid and “re-

duced” projection operators Pai

sidstid sZurek, 1993d.

5. Schmidt states vs pointer basis as projectors

The ability of the instantaneous eigenstates of the re-duced density matrix sSchmidt states; see also Sec.III.E.4d to serve as projectors for consistent histories andpossibly to lead to the emergence of quasiclassical histo-ries has been studied in much detail sAlbrecht, 1992,1993; Paz and Zurek, 1993; Zurek, 1993; Kent andMcElwaine, 1997d. Paz and Zurek s1993d have shown

that Schmidt projectors Pai

sid, defined by their commuta-tivity with the evolved, path-projected reduced densitymatrix,

fPai

sid,Usti−1,tidh¯Ust1,t2dPa1

s1drSst1dPa1

s1dU†st1,t2d ¯ j

3U†sti−1,tidg = 0, s4.16d

will always give rise to an infinite number of sets of con-sistent histories s“Schmidt histories”d. However, thesehistories are branch dependent fsee Eq. s4.7dg, and usu-ally extremely unstable, since small modifications of thetime sequence used for the projections sfor instance, bydeleting a time pointd will typically lead to drasticchanges in the resulting history, indicating that Schmidthistories are usually very nonclassical sPaz and Zurek,1993; Zurek, 1993d.

This situation is changed when the time points ti arechosen such that the intervals sti+1− tid are larger thanthe typical decoherence time tD of the system, overwhich the reduced density matrix becomes approxi-mately diagonal in the preferred pointer basis chosenthrough environment-induced superselection ssee alsothe discussion in Sec. III.E.4d. When the resultingpointer states, rather than the instantaneous Schmidtstates, are used to define the projection operators, stablequasiclassical histories will typically emerge sPaz andZurek, 1993; Zurek, 1993d. In this sense, it has been sug-gested that interaction with the environment can pro-vide the missing selection criterion that ensures the qua-siclassicality of histories, i.e., their stabilityspredictabilityd, and the correspondence of the projec-tion operators sthe pointer basisd to the preferred deter-minate quantities of our experience.

The approximate noninterference, and thus consis-tency, of histories based on local density operators sen-ergy, number, momentum, charge, etc.d as quasiclassicalprojectors sthe so-called hydrodynamic observables;Gell-Mann and Hartle, 1991b; Dowker and Halliwell,1992; Halliwell, 1998d has been attributed to the dynami-cal stability exhibited by the eigenstates of the local den-sity operators. This stability leads to decoherence in thecorresponding basis sHalliwell, 1998, 1999d. It has beenargued by Zurek s2003ad that this behavior and thus thespecial quasiclassical properties of hydrodynamic ob-servables can be explained by the fact that these observ-ables obey the commutativity criterion, Eq. s3.21d, of theenvironment-induced superselection approach.



6. Exact vs approximate consistency

In the idealized case where the pointer states lead toan exact diagonalization of the reduced density matrix,histories composed of the corresponding pointer projec-tors will automatically be consistent. However, under re-alistic circumstances decoherence will typically lead onlyto approximate diagonality in the pointer basis. This im-plies that the consistency criterion will not be fulfilledexactly and that hence the probability sum rules willonly hold approximately—although usually, due to theefficiency of decoherence, to a very good approximationsGriffiths, 1984; Gell-Mann and Hartle, 1991b; Albrecht,1992, 1993; Omnès, 1992, 1994; Paz and Zurek, 1993;Twamley, 1993b; Zurek, 1993d. Hence, the consistencycriterion has been viewed both as overly restrictive,since the quasiclassical pointer projectors rarely obeythe consistency equations exactly, and as insufficient, be-cause it does not give rise to constraints that wouldsingle out quasiclassical histories.

A relaxation of the consistency criterion has thereforebeen suggested, leading to “approximately consistenthistories” whose decoherence functional would be al-lowed to contain nonvanishing off-diagonal terms scor-responding to a violation of the probability sum rulesd aslong as the net effect of all the off-diagonal terms was“small” in the sense of remaining below the experimen-tally detectable level ssee, for example, Gell-Mann andHartle, 1991b; Dowker and Halliwell, 1992d. Gell-Mannand Hartle s1991bd have even ascribed a fundamentalrole to such approximately consistent histories, a movethat has sparked much controversy and has been consid-ered as unnecessary and irrelevant by some sDowkerand Kent, 1995, 1996d. Indeed, if only approximate con-sistency is demanded, it is difficult to regard this condi-tion as a fundamental concept of a physical theory, andthe question of how much consistency is required willinevitably arise.

7. Consistency and environment-induced superselection

The relationship between consistency andenvironment-induced superselection, and therefore theconnection between the decoherence functional and thediagonalization of the reduced density matrix throughenviornmental decoherence, has been investigated byvarious authors. The basic idea, promoted, for example,by Zurek s1993d and Paz and Zurek s1993d, is to suggestthat if the interaction with the environment leads torapid superselection of a preferred basis, which approxi-mately diagonalizes the local density matrix, coarse-grained histories defined in this basis will automaticallybe sapproximatelyd consistent.

This approach has been explored by Twamley s1993bd,who carried out detailed calculations in the context of aquantum-optical model of phase-space decoherence andcompared the results with two-time projected phase-space histories of the same model system. It was foundthat when the parameters of the interacting environmentwere changed such that the degree of diagonality of thereduced density matrix in the emerging preferred

pointer basis was increased, histories in that basis alsobecame more consistent. For a similar model, however,Twamley s1993ad also showed that consistency and di-agonality in a pointer basis as possible criteria for theemergence of quasiclassicality may exhibit a very differ-ent dependence on the initial conditions.

Extensive studies on the connection between Schmidtstates, pointer states, and consistent quasiclassical histo-ries have also been made by Albrecht s1992, 1993d,based on analytical calculations and numerical simula-tions of toy models for decoherence, including detailednumerical results on the violation of the sum rule forhistories composed of different sSchmidt and pointerdprojector bases. It was demonstrated that the presenceof stable system-environment correlations s“records”d,as demanded by the criterion for the selection of thepointer basis, was of great importance in making certainhistories consistent. The relevance of “records” for theconsistent-histories approach in ensuring the “perma-nence of the past” has also been emphasized by otherauthors, for example, by Paz and Zurek s1993d andZurek s1993, 2003ad, and in the “strong-decoherence”criterion by Gell-Mann and Hartle s1998d. The redun-dancy with which information about the system is re-corded in the environment and can thus be found out bydifferent observers without measurably disturbing thesystem itself has been suggested to allow for the notionof “objectively existing histories,” based onenvironment-selected projectors that represent se-quences of “objectively existing” quasiclassical eventssPaz and Zurek, 1993; Zurek, 1993, 2003a, 2004bd.

In general, damping of quantum coherence caused bydecoherence will necessarily lead to a loss of quantuminterference between individual histories sbut not viceversa—see the discussion by Twamley, 1993bd, since thefinal trace operation over the environment in the deco-herence functional will make the off-diagonal elementsvery small due to the decoherence-induced approximatemutual orthogonality of the environmental states.Finkelstein s1993d has used this observation to propose anew decoherence condition that coincides with the origi-nal definition, Eqs. s4.10d and s4.11d, except for restrict-ing the trace to E, rather than tracing over both S and E.It was shown that this condition not only implies theconsistency condition of Eq. s4.15d, but also character-izes those histories that decohere due to interaction withthe environment and that lead to the formation of“records” of the state of the system in the environment.

8. Summary and discussion

The core difficulty associated with the consistent-histories approach has been to explain the emergence ofthe classical world of our experience from the underly-ing quantum nature. Initially, it was hoped that classical-ity could be derived from the consistency criterionalone. Soon, however, the status and the role of this cri-terion in the formalism and its proper interpretation be-came rather unclear and diffuse, since exact consistencywas shown to provide neither a necessary nor a sufficient



condition for the selection of quasiclassical histories.The inclusion of decoherence effects in the consistent-

histories approach, leading to the emergence of stablequasiclassical pointer states, has been found to yield ahighly efficient mechanism and a sensitive criterion forsingling out quasiclassical observables that simulta-neously fulfill the consistency criterion to a very goodapproximation due to the suppression of quantum co-herence in the state of the system. The central questionis then: What is the meaning and the remaining role ofan explicit consistency criterion in the light of such“natural” mechanisms for the decoherence of histories?Can one dispose of this criterion as a key element of thefundamental theory by noting that for all “realistic” his-tories consistency will be likely to arise naturally fromenvironment-induced decoherence alone?

The answer to this question may actually depend onthe viewpoint one takes with respect to the aim of theconsistent-histories approach. As we have noted before,the original goal was simply to provide a formalism inwhich one could, in a measurement-free context, assignprobabilities to certain sequences of quantum eventswithout logical inconsistencies. The more recent andrather opposite aim would be to provide a formalismthat selects only a very small subset of “meaningful”quasiclassical histories, all of which are consistent withour world of experience, and whose projectors can bedirectly interpreted as objective physical events.

The consideration of decoherence effects that can giverise to effective superselection of possible quasiclassicalsand consistentd histories certainly falls into the lattercategory. It is interesting to note that this approach hasalso led to a departure from the original “closed systemsonly” view to the study of local open quantum systems,and thus to the decomposition of the total Hilbert spaceinto subsystems, within the consistent-histories formal-ism. Besides the fact that decoherence intrinsically re-quires the openness of systems, this move might alsoreflect the insight that the notion of classicality itself canbe viewed as only arising from a conceptual division ofthe universe into parts ssee the discussion in Sec. III.Ad.

Therefore environment-induced decoherence and su-perselection have played a remarkable role inconsistent-histories interpretations: a practical one bysuggesting a physical selection mechanism for quasiclas-sical histories; and a conceptual one by contributing to ashift in our view of originally rather fundamental con-cepts, such as consistency, and of the aims of theconsistent-histories approach, like the focus on descrip-tion of closed systems.

V. CONCLUDING REMARKS

We have presented an extensive discussion of the roleof decoherence in the foundations of quantum mechan-ics, with a particular focus on the implications of deco-herence for the measurement problem in the context ofvarious interpretations of quantum mechanics.

A key achievement of the decoherence program is therecognition that openness in quantum systems is impor-

tant for their realistic description. The well-known phe-nomenon of quantum entanglement had already, early inthe history of quantum mechanics, demonstrated thatcorrelations between systems can lead to “paradoxical”properties of the composite system that cannot be com-posed from the properties of the individual systems.Nonetheless, the viewpoint of classical physics that theidealization of isolated systems is necessary to arrive atan “exact description” of physical systems has influencedquantum theory for a long time. It is the great merit ofthe decoherence program to have emphasized the ubiq-uity and essential inescapability of system-environmentcorrelations and to have established the important roleof such correlations as factors in the emergence of “clas-sicality” from quantum systems. Decoherence also pro-vides a realistic physical modeling and a generalizationof the quantum measurement process, thus enhancingthe “black-box” view of measurements in the standards“orthodox”d interpretation and challenging the postu-late of fundamentally classical measuring devices in theCopenhagen interpretation.

With respect to the preferred-basis problem of quan-tum measurement, decoherence provides a very promis-ing definition of preferred pointer states via a physicallymeaningful requirement, namely, the robustness crite-rion, and it describes methods for selecting operationallysuch states, for example, via the commutativity criterionor by extremizing an appropriate measure such as purityor von Neumann entropy. In particular, the fact thatmacroscopic systems virtually always decohere into po-sition eigenstates gives a physical explanation for whyposition is the ubiquitous determinate property of theworld of our experience.

We have argued that, within the standard interpreta-tion of quantum mechanics, decoherence cannot solvethe problem of definite outcomes in quantum measure-ment: We are still left with a multitude of salbeit indi-vidually well-localized quasiclassicald components of thewave function, and we need to supplement or otherwiseto interpret this situation in order to explain why andhow single outcomes are perceived. Accordingly, wehave discussed how environment-induced superselectionof quasiclassical pointer states together with the localsuppression of interference terms can be put to great usein physically motivating, or potentially disproving, rulesand assumptions of alternative interpretive approachesthat change sor altogether abandond the strict orthodoxeigenvalue-eigenstate link and/or modify the unitary dy-namics to account for the perception of definite out-comes and classicality in general. For example, to namejust a few applications, decoherence can provide a uni-versal criterion for the selection of the branches inrelative-state interpretations and a physical argumentfor the noninterference between these branches fromthe point of view of an observer; in modal interpreta-tions, it can be used to specify empirically adequate setsof properties that can be ascribed to systems; in collapsemodels, the free parameters sand possibly even the na-ture of the reduction mechanism itselfd might be deriv-able from environmental interactions; decoherence can



also assist in the selection of quasiclassical particle tra-jectories in Bohmian mechanics, and it can serve as anefficient mechanism for singling out quasiclassical histo-ries in the consistent-histories approach. Moreover, ithas become clear that decoherence can ensure the em-pirical adequacy and thus empirical equivalence of dif-ferent interpretive approaches, which has led some tothe claim that the choice, for example, between the or-thodox and the Everett interpretation becomes “purelya matter of taste, roughly equivalent to whether one be-lieves mathematical language or human language to bemore fundamental” sTegmark, 1998, p. 855d.

It is fair to say that the decoherence program shedsnew light on many foundational aspects of quantum me-chanics. It paves a physics-based path towards motivat-ing solutions to the measurement problem; it imposesconstraints on the strands of interpretations that seeksuch a solution and thus makes them also more andmore similar to each other. Decoherence remains an on-going field of intense research, in both the theoreticaland experimental domain, and we can expect further im-plications for the foundations of quantum mechanicsfrom such studies in the near future.

ACKNOWLEDGMENTS

The author would like to thank A. Fine for manyvaluable discussions and comments throughout the pro-cess of writing this article. He gratefully acknowledgesthoughtful and extensive feedback on this manuscriptfrom S. L. Adler, V. Chaloupka, H.-D. Zeh, and W. H.Zurek.

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