PRL 95, 200405 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending11 NOVEMBER 2005

Pre- and Post-Selection Paradoxes and Contextuality in Quantum Mechanics

M. S. Leifer and Robert W. SpekkensPerimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada

(Received 12 December 2004; published 11 November 2005)

0031-9007=

Many seemingly paradoxical effects are known in the predictions for outcomes of intermediatemeasurements made on pre- and post-selected quantum systems. Despite appearances, these effects donot demonstrate the impossibility of a noncontextual hidden variable theory, since an explanation in termsof measurement disturbance is possible. Nonetheless, we show that for every paradoxical effect whereinall the pre- and post-selected probabilities are 0 or 1 and the pre- and post-selected states are non-orthogonal, there is an associated proof of the impossibility of a noncontextual hidden variable theory.This proof is obtained by considering all the measurements involved in the paradoxical effect—thepreselection, the post-selection, and the alternative possible intermediate measurements—as alternativepossible measurements at a single time.

DOI: 10.1103/PhysRevLett.95.200405 PACS numbers: 03.65.Ta

The study of quantum systems that are both pre- andpost-selected was initiated by Aharonov, Bergmann, andLebowitz (ABL) in 1964 [1], and has led to the discoveryof many counter-intuitive results, which we refer to as pre-and post-selection (PPS) effects [2], some of which haverecently been confirmed experimentally [3].

These results have led to a long debate about the inter-pretation of the ABL probability rule [4]. An undercurrentin this debate has been the connection between PPS effectsand contextuality. As first demonstrated by Albert,Aharonov, and D’Amato [5], using an effect now knownas the ‘‘3-box paradox,’’ the probability assigned to an out-come of a measurement given a pre- and a post-selectioncan depend not only on the projector associated with thatoutcome, but also on other details of the observable. This isa sort of context dependence. As such, it seems to under-mine the significance of the Bell-Kochen-Specker theorem[6], which establishes the impossibility of a noncontextualhidden variable theory (HVT), since there is no reason tothink that probabilities conditioned on the hidden variablesshould be noncontextual if quantum probabilities are them-selves contextual. Bub and Brown [7] convincingly dis-puted this argument by giving an example for which simi-lar reasoning would imply that quantum theory allowedsuperluminal signalling. As they showed, it is a mistake tothink that a pre- and post-selected ensemble can be definedindependently of the intermediate measurement. Althoughwe agree that PPS paradoxes are not themselves proofs ofthe contextuality of HVTs, we show that there is nonethe-less a close connection between the two.

This connection is expected to have interesting applica-tions in quantum foundational studies. For instance, it hasbeen suggested that Bell’s theorem [8] might be under-stood within a realist and Lorentz-invariant framework ifone admits the possibility of a HVT that allows forbackward-in-time causation [9]. A simple model haseven been suggested by Kent [10]. This is closely con-nected to the fact that Bell correlations can be simulated

05=95(20)=200405(4)$23.00 20040

using post-selection, as shown in Ref. [7], which in turn isthe root of the detection-efficiency loophole in experimen-tal tests of Bell’s theorem [11]. Further investigations intothe connection between proofs of nonlocality and PPSparadoxes would shed new light on these avenues of re-search. As nonlocality is a kind of contextuality (assumingseparability [12]), the connection between contextualityand PPS paradoxes established in the present work is animportant contribution to this project.

Mermin [13] has already shown one connection betweenPPS effects and contextuality. His investigation concernedwhat is known as the ‘‘mean king’s problem’’ which is aPPS effect that is qualitatively different from the paradoxi-cal variety of PPS effect that we shall be considering.Moreover, Mermin demonstrated how one can obtain atype of mean king’s problem that is unsolvable startingfrom the measurements used in a proof of the contextualityof HVTs, whereas we demonstrate how one can obtainproofs of contextuality starting from the measurementsused in a PPS paradox.

To be specific, we show that for every PPS paradoxwherein all the PPS probabilities are 0 or 1 and the pre-and post-selection states are nonorthogonal, there is anassociated proof of the contextuality of HVTs. The keyto the proof is that measurements that are treated as tem-poral successors in the PPS paradox are treated as counter-factual alternatives in the proof of contextuality. This resultsuggests the existence of a subtle conceptual connectionbetween the two phenomena that has yet to be fully under-stood. Thus, the present work contributes to the project ofreducing the number of logically distinct quantum mys-teries by revealing the connections between them.

We begin with a curious prediction of the ABL ruleknown as the 3-box paradox. Consider a particle that canbe in one of three boxes. Denote the state where the particleis in box j by jji. The particle is preselected in the statej�i � j1i � j2i � j3i and post-selected in the state j i �j1i � j2i � j3i (states will be left unnormalized). At an

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PRL 95, 200405 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending11 NOVEMBER 2005

intermediate time, one of two possible measurements isperformed. The first possibility corresponds to the projec-tor valued measure (PVM) [14] E1 � fP1; P?1 g, whereP1 � j1ih1j and P?1 � j2ih2j � j3ih3j. The second possi-bility corresponds to the PVM E2 � fP2; P

?2 g, where P2 �

j2ih2j and P?2 � j1ih1j � j3ih3j.Now note that P?1 can also be decomposed into a sum of

the projectors onto the vectors j2i � j3i and j2i � j3i. Thefirst of these is orthogonal to the post-selected state, whilethe second is orthogonal to the preselected state, so theprobability of the outcome P?1 occurring, given that thepre- and post-selection were successful, must be 0. Conse-quently, the measurement of E1 necessarily has the out-come P1. Similarly, P?2 can be decomposed into a sum ofthe projectors onto the vectors j1i�j3i and j1i�j3i, whichare also orthogonal to the post- and preselected states, re-spectively. Consequently, the measurement of E2 neces-sarily has the outcome P2. Thus, if one measures to seewhether or not the particle was in box 1, one finds that itwas in box 1 with certainty, and if one measures to seewhether or not it was in box 2, one finds that it was in box 2with certainty.

This is reminiscent of the sort of conclusion that oneobtains in proofs of the impossibility of a noncontextualhidden variable theory. Indeed, a proof presented byClifton [15] makes use of the same mathematical structure,as we presently demonstrate.

Consider the eight vectors mentioned in our discussionof the 3-box paradox, but imagine that these describealternative possible measurements at a single time (incontrast to what occurs in the 3-box paradox). In a non-contextual HVT, it is presumed that although not all ofthese tests can be implemented simultaneously, their out-comes are determined by the values of preexisting hiddenvariables and are independent of the manner in which thetest is made (the context). Thus each of these vectors isassigned a value, 1 or 0, specifying whether the associatedtest would be passed or not. Clearly, the following tworules must apply: (1) For any orthogonal pair, not both canreceive the value 1, and (2) for any orthogonal triplet,exactly one must receive the value 1. Representing thevectors by points and orthogonality by lines, the eightvectors can be depicted as in Fig. 1.

Clifton’s proof is an example of a probablistic proof ofthe contextuality of HVTs, since it relies on assigning thevectors j�i; j i probability 1 a priori. This is justified as

|2 − |3 |2 + |3

|1

|2|1 + |2 + |3 |1 + |2 − |3

|1 − |3 |1 + |3

|2⟩ − |3⟩ |2⟩ + |3⟩

|1⟩

|2⟩|1⟩ + |2⟩ + |3⟩ |1⟩ + |2⟩ − |3⟩

|1⟩ − |3⟩ |1⟩ + |3⟩

FIG. 1. The vectors in Clifton’s proof of contextuality. White(black) represents the value 1 (0).

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follows: the state j�i can be prepared, and if it is, then asubsequent test for j�i will be passed with certainty. Thus,for all valuations of the hidden variables consistent with thepreparation of j�i; j�i is assigned the value 1. Further-more, after a preparation of j�i a test for j iwill be passedwith nonzero probability (because h j�i � 0) and conse-quently some of the valuations of the hidden variablesconsistent with the preparation of j�i also assign value 1to j i. Let � denote a valuation that assigns 1 to both j�iand j i. Since j1i � j3i and j2i � j3i are orthogonal toj�i, they must be assigned value 0 by � and since j1i � j3iand j2i � j3i are orthogonal to j i, they must also beassigned value 0 by �. But given that j1i, j2i � j3i, andj2i � j3i form an orthogonal triplet, by rule (2) it followsthat j1i must be assigned the value 1 by �. Similarly, giventhat j2i, j1i � j3i, and j1i � j3i form an orthogonal triplet,by rule (2) it follows that j2i must be assigned the value 1by �. However, by rule (1) j1i and j2i cannot both receivethe value 1, since they are orthogonal. Thus, we havederived a contradiction.

To our knowledge, the connection between Clifton’sproof and the 3-box paradox has not previously beenrecognized.

We will show that this sort of connection is generic toPPS paradoxes. We begin with a short review of the ABLrule, hidden variable theories, and contextuality.

We only consider quantum systems with a finite dimen-sional Hilbert space and assume that no evolution occursbetween measurements. We restrict our attention to sharpmeasurements, that is, those associated with PVMs. Wealso restrict attention to measurements for which the stateupdates according to �! Pj�Pj=Tr�Pj�� upon obtainingoutcome j. This is known as the Luders rule. We call thisset of assumptions the standard framework for PPS effects.It includes all of the PPS ‘‘paradoxes’’ discussed in theliterature to date.

Now, consider a temporal sequence of three sharp mea-surements. The initial, intermediate, and final measure-ments occur at times tpre, t, and tpost respectively. Theonly relevant aspects of the initial and final PVMs are theprojectors associated with the outcomes specified by thepre- and post-selection. We denote these by �pre and �post,and we denote the PVM associated with the intermediatemeasurement by E � fPjg.

Assuming that nothing is known about the system priorto tpre, so that the initial density operator is I=Tr�I�, where Iis the identity operator, the measurement at tpre preparesthe density operator �pre � �pre=Tr��pre�. By Bayes’theorem, we can deduce that the probability of obtainingthe outcome k in the intermediate measurement is

p�Pkj�pre;�post; E� �Tr��postPk�prePk�P

jTr��postPj�prePj�

: (1)

This is a special case of the most general version of theABL rule [2], and we therefore refer to such probabilities

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PRL 95, 200405 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending11 NOVEMBER 2005

as ‘‘ABL probabilities.’’ In the case where �pre and �post

are rank-1 projectors onto states j�i and j i respectively,this rule reduces to the more familiar p�Pkj�; ; E� �jh jPkj�ij

2=Pjjh jPjj�ij

2 which was implicitly used inour discussion of the 3-box paradox.

A hidden variable theory is an attempt to explain theoutcomes of quantum measurements by a set of variablesthat are hidden to one who knows only the preparationprocedure. We use the term ontic state to refer to a com-plete description of the real state of affairs according to theHVT. This includes a valuation of all the hidden variables,and may also include the quantum state vector if this hasontic (rather than epistemic) status in the HVT. A particu-larly natural class of HVTs are those that satisfy thefollowing two assumptions [12]: measurement noncontex-tuality, which is the assumption that the manner in whichthe measurement is represented in the HVT depends onlyon the PVM and not on any other details of the measure-ment (the context); and outcome determinism for sharpmeasurements, which is the assumption that the outcome ofa PVM measurement is uniquely fixed by the ontic state.We abbreviate these as MNHVTs. It follows that in anMNHVT, every projector is assigned a probability of 0 or 1by an ontic state, independently of the nature of the mea-surement in which the projector appears, for instance, ofwhat other projectors might be measured simultaneouslywith it. In other words, projectors are associated withunique preexisting properties that are simply revealed bymeasurement [12].

Suppose we denote by s the proposition that asserts thatthe property associated with projector P is possessed. In anMNHVT the negation of s, denoted :s, is associated withI � P. Now consider a projector Q that commutes with P,and denote the proposition associated with Q by t. In anMNHVT the conjunction of s and t, denoted s ^ t, isassociated with PQ and the disjunction of s and t, denoteds _ t, is associated with P�Q� PQ (the latter followsfrom the fact that s _ t � :�:s ^ :t��.

Let p�s� denote the probability that the proposition s istrue. Classical probability theory dictates that 0 � p�s� �1, p�:s� � 1� p�s�, p�s _ :s� � 1, p�s ^ :s� � 0,p�s ^ t� � p�s�, p�s ^ t� � p�t�, and p�s _ t� �p�s� � p�t� � p�s ^ t�. We therefore obtain the followingconstraints on an MNHVT.

Algebraic conditions.—For projectors P;Q such that�P;Q� � 0, we have 0 � p�P� � 1, p�I � P� �1� p�P�, p�I� � 1, p�0� � 0, p�PQ� � p�P�, p�PQ� �p�Q�, and p�P�Q� PQ� � p�P� � p�Q� � p�PQ�.

The Bell-Kochen-Specker theorem shows that there aresets of projectors to which no assignment of probabilities 0or 1 consistent with the algebraic conditions is possible.This demonstrates the impossibility of an MNHVT, orequivalently, the contextuality of HVTs of quantum me-chanics [16].

A connection to PPS paradoxes is suggested by the factthat there exist sets of projectors for which an ABL proba-

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bility assignment violates the algebraic constraints, whileevery projector receives probability 0 or 1. We call such ascenario a logical PPS paradox. The 3-box paradox is anexample of this [17].

Now, define � to be the set (or ensemble) of ontic statesthat can obtain prior to the intermediate measurementgiven the pre- and post-selection. If � were independentof the nature of the intermediate measurement, then theprobability assigned by the ABL rule to a projector couldalso be interpreted as the probability assigned to it by onticstates in �. But since the latter probabilities are required tosatisfy the algebraic conditions in an MNHVT, the viola-tion of these conditions would be a proof of the impossi-bility of an MNHVT. However, a measurement in an HVTneed not be modeled simply by a Bayesian updating ofone’s information, but may also lead to a disturbance of theontic state. Thus, to determine � we need to work back-wards from the post-selection, asking what is the set � ofontic states that could obtain after the intermediate mea-surement given the post-selection, and then what onticstates could obtain prior to the intermediate measurementgiven � and given the nature of the disturbance. Since thedisturbance may depend on the nature of the intermediatemeasurement, so too may �. Thus, the possibility ofmeasurement disturbance blocks the conclusion that aPPS paradox is itself a proof of the contextuality ofHVTs. This is discussed in more detail in Ref. [20].

Despite these considerations, the main aim of this letteris to show that there is a connection between PPS para-doxes and contextuality, but it is significantly more subtlethan one might have thought.

Theorem.—For every logical PPS paradox within thestandard framework for which the pre- and post-selectionprojectors are nonorthogonal, there is an associated proofof the impossibility of an MNHVT that is obtained byconsidering all the measurements defined by the PPS para-dox—the preselection, the post-selection, and the alterna-tive possible intermediate measurements—as alternativepossible measurements at a single time.

Our proof of this theorem generalizes the argumentpresented for the 3-box paradox. We begin with two lem-mas and a corollary.

Lemma 1.—If �pre, �post, P are projectors satisfying�post�I � P��pre � 0, then there exists a pair of orthogo-nal projectors Q and R such that I � P � Q� R where�preR � 0 and �postQ � 0.

Proof.—Let R �I � P� ^ �I ��pre�, where P ^Qdenotes the projector onto the intersection of the subspacesassociated with P and Q. This clearly satisfies �preR � 0.Moreover, since R is a subspace of I � P, the projectorQ �I � P� � R is orthogonal to R and satisfies I � P �Q� R. Finally, �post�I � P��pre � 0 entails that �post isorthogonal to the projector onto ran��I � P��pre�, whereran�X� denotes the range of X. But this projector is simply�I � P� � �I � P� ^ �I ��pre� � Q. Thus, �postQ � 0 issatisfied. �

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Lemma 2.—If under a preselection of �pre and a post-selection of �post, the projector P receives probability 1 ina measurement of a PVM E � fP;P?1 ; P

?2 ; . . .g that appears

in a logical PPS paradox, then in an MNHVT, if �pre and�post are assigned probability 1 by some ontic state �, P isalso assigned probability 1 by the ontic state �. Succinctly,if p�Pj�pre;�post; E� � 1 and p��prej�� � p��postj�� �1, then p�Pj�� � 1.

Proof.—If p�Pj�pre;�post; E� � 1, it follows from theABL rule that Tr��preP?k �postP?k � � 0 for all k. SinceTr�AyA� � 0 implies that A � 0; it follows that�postP

?k �pre � 0 for all k and consequently that �post�I �

P��pre � 0. It then follows from lemma 1 that I � P canbe decomposed into a sum of projectors R andQ which areorthogonal to �pre and �post respectively. Given this or-thogonality, for any � in an MNHVT yielding p��prej�� �p��postj�� � 1, we have p�Qj�� � p�Rj�� � 0. The al-gebraic conditions then imply that p�Pj�� � 1�p�I � Pj�� � 1� p�Q� Rj�� � 1. �

Corollary.—If p�Pj�pre;�post; E� � 0 and p��prej�� �p��postj�� � 1, then p�Pj�� � 0.

Proof.—Given that each of the projectors P;P?1 ;P?2 ; . . . must receive value 0 or 1 in a logical PPSparadox, p�Pj�pre;�post; E� � 0 implies thatp�P?k j�pre;�post; E� � 1 for some k, which by lemma 2implies that p�P?k j�� � 1 for some k. It then follows fromthe algebraic constraints that p�Pj�� � 0. �

Proof of theorem.—By the assumption that �pre and�post are nonorthogonal, there exist ontic states � suchthat p��prej�� � p��postj�� � 1. This, together withLemma 2 and its corollary, implies that whatever proba-bility assignments to fPg arise from the ABL rule also arisein any MNHVT as the probability assignment to fPg forsuch �. Since, by the assumption of a logical PPS paradox,the ABL probabilities violate the algebraic conditions, itfollows that the probabilities conditioned on such � in anMNHVT also violate the algebraic conditions. However,probability assignments in an MNHVT must satisfy theseconditions, therefore an MNHVT is ruled out. �

A question that has not been addressed in the presentwork is whether, in an arbitrary statistical theory (not justquantum theory), the existence of logical PPS paradoxesimplies that this theory cannot be modeled by a noncon-textual HVT. To answer this question, one must character-ize PPS paradoxes and contextuality in a theory-independent manner. For an attempt to generalize thenotion of contextuality to HVTs for arbitrary statisticaltheories, see Ref. [12]. No attempt at providing an opera-tional characterization of logical PPS paradoxes has yet

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been made; however, the surprising features of the 3-boxparadox have been reproduced within two simple toytheories, described in Refs. [20,21], and these toy theoriescan be understood in terms of a HVT that is noncontextualby the definition of Ref. [12]. Thus it seems that theexistence of logical PPS paradoxes does not imply theimpossibility of a noncontextual HVT. Nonetheless, theremay be additional natural constraints that a general class oftheories including quantum mechanics satisfy, under whichthis implication holds true. Further investigations intothese issues are required.

We would like to thank Ernesto Galvao, Terry Rudolph,and Alex Wilce for helpful comments.

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[1] Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys.Rev. 134, B1410 (1964).

[2] Y. Aharonov and L. Vaidman, in Time in QuantumMechanics, edited by J. G. Muga, R. Sala Mayato, andI. L. Egusquiza (Springer, New York, 2002), pp. 369–412.

[3] J. Lundeen, K. Resch, and A. Steinberg, Phys. Lett. A 324,125 (2004).

[4] R. E. Kastner, Philosophy of Science 70, 145 (2003).[5] D. Z. Albert, Y. Aharonov, and S. D’Amato, Phys. Rev.

Lett. 54, 5 (1985).[6] J. S. Bell, Rev. Mod. Phys. 38, 447 (1966); S. Kochen and

E. P. Specker, J. Math. Mech. 17, 59 (1967).[7] J. Bub and H. Brown, Phys. Rev. Lett. 56, 2337 (1986).[8] J. S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1964).[9] H. Price, Time’s Arrow and Archimedes’ Point (Oxford

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[11] A. Fine, Found. Phys. 19, 453 (1989).[12] R. W. Spekkens, Phys. Rev. A 71, 052108 (2005).[13] N. D. Mermin, Phys. Rev. Lett. 74, 831 (1995).[14] A PVM is a set of projectors that sum to identity.[15] R. Clifton, Am. J. Phys. 61, 443 (1993).[16] As emphasized by Bell [6], contextual HVTs are possible,

Bohmian mechanics being a prime example.[17] Another example is the ‘‘failure of the product rule’’

discussed in Ref. [18]. By our results, this is related tothe contextuality proofs discussed in Ref. [19].

[18] L. Vaidman, Phys. Rev. Lett. 70, 3369 (1993); O. Cohen,Phys. Rev. A 51, 4373 (1995).

[19] N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990); A. Peres,J. Phys. A 24, L175 (1991); M. Kernaghan, J. Phys. A 27,L829 (1994).

[20] M. S. Leifer and R. W. Spekkens, quant-ph/0412179 [Int.J. Theor. Phys. (to be published)].

[21] K. A. Kirkpatrick, J. Phys. A 36, 4891 (2003).