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ROBIN HUDSONS PATHLESS PATH TO QUANTUM

STOCHASTIC CALCULUS

DAVID APPLEBAUM

Abstract. Robin Hudsons work on quantum central limit theorems, quan-

tum Brownian motion, quantum stopping times and formal quantum sto-chastic calculus is reviewed and reappraised.

1. Introduction

Robin Hudson and K. R. Parthasarathy are the creators of quantum stochasticcalculus. The key paper which contained almost all the basic results was publishedin 1984 in Communications in Mathematical Physics [19]. Here we find, inparticular, the construction of quantum stochastic integrals, quantum Itos formula,the existence and uniqueness of linear quantum stochastic differential equations(QSDEs), necessary and sufficient conditions for unitarity of solutions and thedilation of quantum dynamical semigroups (at least for one degree of freedom). Inthis article I will focus on the pre-history of quantum stochastic calculus with aparticular emphasis on Robin Hudsons contribution. This will cover work that waspublished in the period 1971-84. As can be seen by a quick journey to MathSciNet,this was a highly productive period for Robin. I am not going to survey all hispapers from this period in this article or even all of those that he wrote in the area

of quantum probability. What I will present is four key milestones - the quantumcentral limit theorem, quantum Brownian motion, quantum stopping times andthe heuristic version of quantum stochastic calculus that preceded its rigorousdevelopment in [19].

A brief comment on the title. In the ancient Chinese philosophy of Taoism, themysterious tao is often described (at least in contemporary English translation) asa pathless path. Of course in quantum theory, a particle does not have a pathin the usual sense and consequently path space techniques are inappropriate toolsfor studying quantum processes.

Notation. IfH is a complex Hilbert space then B(H) is the algebra of allbounded linear operators on H. If T is a densely defined closeable linear operatordefined on H with adjoint T, then any proposition involving T# should be readtwice, once for T and once for T.

Received 2010-4-12; Communicated by the editors.2000 Mathematics Subject Classification. Primary 81S25; Secondary 81Q12, 81S05, 81R15,

81R30.Key words and phrases. Quantum central limit theorem, quantum Brownian motion, quan-

tum stop times, Fock space, quantum stochastic calculus, quantum diffusions.

481

Serials Publicationswww.serialspublications.com

Communications on Stochastic AnalysisVol. 4, No. 4 (2010) 481-491

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482 DAVID APPLEBAUM

2. Quantum Central Limit Theorem

Robins first paper on quantum probability was joint work with his PhD student

Clive Cushen [9]. The opening words of the introduction to this paper are almost aclarion call for quantum probability: In recent years there has been an increasingawareness that the foundations of quantum mechanics lie in a non-commutativeanalogue of axiomatic probability theory. In order to formulate a quantum centrallimit theorem (CLT), Cushen and Hudson first needed to decide what should aquantum random variable be and how could a sequence of these be identicallydistributed and independent? The basic (bosonic) quantum random variable isa canonical pair (q, p) of linear self-adjoint operators acting in a complex Hilbertspace H and satisfying the Heisenberg commutation relation

[p, q] := pq qp = iI,on a suitable dense domain. The distribution of the pair (q, p) is determined bya mixed state which is identified with its density operator . Now consider an

infinite sequence ((qn, pn), n N) of such canonical pairs satisfying[qi, qj ] = [pi, pj ] = 0; [pi, qj ] = iijI,

for each i, j N and equipped with the state . They are said to be independentif all finite subsets are independent in the sense that if A N is finite withA = {i1, . . . , iN} then A is unitarily equivalent to i1 iN. Here A is thereduced density operator defined on L2(RN) by

tr(AT) = tr((T)),

for all T B(L2(RN)) and is the canonical embedding of B(L2(RN)) into B(H)that is determined by von Neumanns uniqueness theorem. The reduced statesi1 , . . . , iN are similarly obtained by taking A = {i1}, . . . ,{iN} (respectively).The sequence ((qn, pn), n N) comprises identically distributed quantum randomvariables if i1 = = iN for every finite set A

N. Now we average. For eachn N, define

pn =1

n(p1 + +pn), qn =

1

n(q1 + + qn).

It is easy to see that (qn, pn) form a canonical pair and the main result of thepaper is to prove the quantum central limit theorem:

limn

tr(nT) = tr(T), (2.1)

for all T B(L2(R)). Here n is the reduced density operator correspondingto the canonical pair (qn, pn) and is a quantum Gaussian state on L

2(R), i.e.a thermal state of the quantum harmonic oscillator having variance 1 (seeExample 2 in [3] for insight into the sense in which this state is Gaussian and[28] for an expository account of quantum Gaussian states.)

A key ingredient in the proof is the use of quasi-characteristic functions, whichare defined for x, y R byfp,q(x, y) = tr(Ux,y),

where U(x, y) = ei(xp+yq) is the Weyl operator. Indeed the authors establish aGlivenko-type convergence theorem to the effect that for (qn, pn) to converge in

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ROBIN HUDSONS PATHLESS PATH 483

distribution (i.e. in the sense of 2.1) it is sufficient for the associated sequenceof quasi-characteristic functions to converge pointwise to a function on R2 that iscontinuous at the origin.

This paper was followed by the fermionic version [13]. In this case, the ap-propriate analogue of the canonical pairs are fermionic annihilation and creationoperators (an, a

n), n N) which satisfy the canonical anticommutation relations

(CARs):

{ai, aj} = {ai , aj} = 0; {ai, aj} = ijI,for i, j N, where {A, B} := AB + BA is the anticommutator. Once again weget a new representation of the CARs by averaging:

a#n :=1

n(a#1 + + a#n ),

and the fermionic central limit theorem yields convergence of corresponding re-duced states to a fermionic quasi-free state (i.e. a fermionic Gaussian). In this

work, the key tool was cumulants rather than quasi-characteristic functions. Thedevelopment of the appropriate fermionic version of the latter required the use ofGrassman algebra techniques [2].

The Cushen-Hudson work is without doubt a landmark paper, and not just forits influence on quantum probability. The study of quantum central limit theoremsis a flourishing enterprise in its own right which has attracted the attention of alarge number of authors since the early 1970s, see e.g. [11], [10], [12] and [21]. In-deed the recent article [21] lists seven distinct areas in which quantum central limittheorems have been developed and applied including quantum information theory,graph theory and combinatorics. K.R.Parthasarathy told the author1 that read-ing this paper for the first time deeply influenced his subsequent mathematicallife (sic.)

3. Quantum Brownian Motion

The first paper to study quantum Brownian motion was published by Robintogether with another of his PhD students2 Anne Cockcroft in 1977 [7]. In thispaper, the passage of time is modelled by the closed interval [0, 1] , but the authorscould just as easily used R+ := [0,) and this became the standard choice in laterwork. A quantum Brownian motion3 is a pair (P(t), Q(t), t [0, 1]) of self-adjointoperator-valued functions acting in a complex Hilbert space H together with adistinguished vector to determine expectations such that:

(i) [P(s), P(t)] = [Q(s), Q(t)] = 0; [P(s), Q(t)] = is tfor all s, t [0, 1].

(ii) P(0) = Q(0) = 0.

1E-mail communication in April 20102Robin has always been extremely generous in sharing his ideas with others and many PhD

students, including the author, have been beneficiaries of this largesse.3In [7] the authors used the terminology quantum Wiener process, but quantum Brownian

motion became the preferred usage amongst practitioners.

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484 DAVID APPLEBAUM

(iii) For := (a, b] [0, 1], define the canonical pair (p, q) by

p :=p(b) p(a)b a

, q :=q(b) q(a)b a

.

For arbitrary pairwise disjoint (n, n N), the sequence ((pn), qn)), n N) consists of independent and identically distributed (i.i.d.) canonicalpairs having quantum Gaussian distributions with mean 0 and variance2 1 in the state determined by .

Note that each of (Q(t), t 0) and (P(t), t 0) are separately equivalent tothe probabilists Brownian motion but the non-trivial commutation relation in (i)ensures that these are not simultaneously diagonisable. The key probabilistic inputof the definition is in (iii) which extends to a non-commutative framework the factthat a classical Brownian motion (B(t), t 0) has stationary and independentincrements with each random variable B(t)B(s)

ts N(0, 2).The main result of this paper is to show that any quantum Brownian motion is

unitarily equivalent to the pair of co-ordinate and momentum field operatorswhich are indexed by the indicator function 1[0,t) and are associated to a cyclicrepresentation of the extremal universally invariant quasi-free state on the WeylCCR algebra over H defined on Weyl operators W(f) by

(W(f)) = e 1

22||f||2

for each f H. The case = 1 is the Fock state.In later years, the development of quantum stochastic calculus, made it more

convenient to identify quantum Brownian motion with the annihilation/creationoperator valued process (A(t), A(t), t 0) defined by

A(t) =1

2(Q(t) + iP(t)), A(t) =

12

(Q(t) iP(t)),

which satisfy the commutation relations (CCRs):[A(s), A(t) = [A(s), A(t)] = 0, [A(s), A(t)] = s tI,

for all s, t 0. The Cockroft-Hudson theory then tells us that every quantumBrownian motion for which = 1 is unitarily equivalent to

(a(1[0,t)), a(1[0,t)), t 0),

where a(), a() are the Fock annihilation and creation operators acting in bosonFock space (L2(R+)) with distinguished vector the Fock vacuum . When >1, quantum Brownian motion is often said to be non-Fock. In this case, thereference Hilbert space is (L2(R+)) (L2(R+)) and we have

A(t) = a(1[0,t)) I + I a(1[0,t)), A(t) = a(1[0,t)) I + I a(1[0,t)),

where2

2

= 1, 2

+2

=2

. The distinguished vector is . The non-Fockquantum Brownian motions have a deep and beautiful mathematical structure(see e.g. [16]) and it may well be that they havent yet been exploited to their fullpotential.

The Cockcroft-Hudson paper is justly celebrated as marking the birth of quan-tum Brownian motion. Perhaps less well-known is the follow-up paper [8] by the

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same authors (but now augmented with S.Gudder) which directly followed it in thesame volume of the Journal of Multivariate Analysis. Here the authors establisha functional central limit theorem for quantum Brownian motion. The set-up isas follows. Suppose that we have a sequence of canonical pairs ((pn, qn), n N)that are i.i.d. with respect to a state for which

(qn) = (pn) = ({pn, qn}) = 0; (q2n) = (p2n) = 2,for all n N, where 2 1. For each t [0, 1] consider the operators defined by

Pn(t) =1n

(p1 + +p[nt] + (nt [nt])p[nt]+1),

Qn(t) =1n

(q1 + + q[nt] + (nt [nt])q[nt]+1).It follows that for each s, t [0, 1]

[Pn(s), Pn(t)] = [Qn(s), Qn(t)] = 0, [Pn(s), Qn(t)] = i(s t + rn),where limn rn = 0. The authors demonstrate that the sequence ((Pn, Qn),n N) converges weakly to quantum Brownian motion of variance 2. A largepart of the paper grapples with the question of what weak convergence mightmean in this context. The authors build an elaborate technical apparatus whichinter alia requires the compact uniform closure of a C-algebra with respect toa sequence of states. Readers who want to know more about this are referred tothe original paper.

The importance of the Cockroft-Hudson paper [7] lies firstly in its identifica-tion of what would become two of the key fundamental noise processes of quan-tum stochastic calculus and secondly in providing a model for latter versions ofnon-commutative Brownian motions appearing in different contexts such as thefermionic [2], free [32], twisted [6] and monotone [26]. There have not beenmany developments in the literature on quantum probabilistic properties of quan-

tum Brownian motion outside its use as noise in quantum stochastic calculus,although a recent paper by the author [3] has established a Levy-Cielsielski typeseries expansion in terms of a Schauder system. On the other hand, classicalBrownian motion remains a topic of intense study for classical probabilists as itcontinues to yield deep and fascinating secrets (see e.g. [24] for an account ofrecent progress.) Is quantum probability missing an opportunity here?

4. Quantum Stop Times

Stopping times play a very important role in classical probability, probabilisticpotential theory and many applications (e.g. consider the problem of pricing anAmerican option in mathematical finance.) In the classic book by Chris Rogersand David Williams there is a wonderful quote from Sid Port that I cant resist

including here (see [30], p.94

): The one thing probabilists can do which analystscant is stop - and they can never forgive us for it.In [14] Robin introduced the concept of a quantum stopping time and proved

the strong Markov property for quantum Brownian motion. Before going on to

4The page reference is to the Cambridge University Press edition.

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486 DAVID APPLEBAUM

describe this it may be worth recalling the classical result. Let (B(t), t 0) bea Brownian motion defined on a probability space (,F, P) and adapted to afiltration (

Ft, t

0). Let T be a stopping time, i.e. a random variable defined on that takes values in [0,] such that the event (T t) Ft for all 0 t < .The strong Markov property asserts that (B(T + t) B(t), t 0) is a Brownianmotion adapted to the filtration (FT+t, t 0) and independent ofFT := {A F, A (T t) Ft for all t 0}.

In [14] Robin works with the quantum Brownian motion (P(t), Q(t)), t 0) ofvariance 2 1 with distinguished state vector . The role of the -algebra isplayed by the von Neumann algebra N := {P(t), Q(t), t 0}, i.e. the smallestvon Neumann algebra containing all the spectral projections of the Ps and Qsand a filtration in this context is the family of increasing sub-algebras (N, 0)where N := {P(t), Q(t), 0 t }. A stopping time T is then a positiveself-adjoint operator having spectral decomposition T =

0

dE() which is such

that E() N for all 0.5 At least formally the random time-shifted quantumBrownian motion should be

PT(t) :=

0

(P(t + ) P())dE(), QT(t) :=

0

(Q(t + ) Q())dE()

for each t 0. In order to give these formal expressions a rigorous meaning, Robindefines them indirectly as infinitesimal generators of the unitary operator-valuedspectral integrals defined for each x R by

UP(t)(x) : =

0

eixP(t+)eixP()dE(),

VQ(t)(x) : =

0

eixQ(t+)eixQ()dE() (4.1)

so (UP(t)(x), x R) and (VQ(t)(x), x R) are each strongly continuous one-parameter unitary groups and we have

UP(t)(x) = eixP(t), VQ(t)(x) = e

ixQ(t),

for each t 0, x R. Much of the technical work in the paper involves theconstruction of integrals of the type considered in (4.1) as limits of Riemann sumsin the strong operator topology.

Before we can state the quantum strong Markov property, we need the conceptsof pre-T and post-T von Neumann algebras which well denote as NT] and N[Trespectively. These are defined by

NT] := {A N, AE() = E()A N for all 0},N[T := {PT(t), QT(t), t 0}.

The strong Markov property states that

NT] and

N[T are independent in the

state 6 and that ((PT(t), QT(t)), t 0) is a quantum Brownian motion of vari-ance 2.

5Later literature on quantum stop times usually defined these directly in terms of projection-valued measures - see e.g. [29].

6i.e. ,AB = ,A,B for all A NT], B N[T.

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A corresponding strong Markov property was established by the author forfermion Brownian motion in [2]. A key later development of quantum stop-ping times was the paper [29] by Parthasarathy and Sinha in which the tenso-rial factorisation of Fock space over L2(R+) corresponding to the splitting f f1[0,t)+f1[t,) was extended to the case where t is replaced by a quantum stoppingtime. Another paper worth mentioning (which sadly has never been followed up inthe literature) is a very interesting study of first exit times by J.-L.Sauvageot [31].Although there has continued to be sporadic work on quantum stopping times (seee.g. [15] for a recent survey article by Robin) it seems that a breakthrough is stillneeded to forge it into a tool that is of similar value in quantum probability to itscommutative counter-part.

5. Formal Quantum Stochastic Calculus

In the last part of this survey I will focus on work carried out during the early1980s. A great deal of the standard conceptual structure of quantum stochastic

calculus was developed by Robin and co-workers (principally K.R.Parthasarathyand R.F.Streater) from a heuristic viewpoint. The rigorous development came alot later. In many ways these formal calculations (which are quite satisfactory tomost physicists) constitute the essence of the subject.

At this time, the basic quantum processes were understood to be the annihila-tion/creation pair (A(t), A(t), t 0) (where A#(t) = a#(1[0,t)) acting in bosonFock space (L2(R+)) and equipped with the vacuum vector to determine ex-pectations. The filtration was induced by the canonical isomorphism between(L2(R+)) and (L2([0, t))) (L2([t,)) which maps each exponential vectore(f) to e(f1[0,t)) e(f1[t,)). In order to define formal quantum stochastic inte-grals we write dA#(t) := a#(1[t,t+dt)). Everything follows from the eigenrela-tion:

A(t)e(f) =t

0 f(s)ds

e(f),

for each f L2(R+). Taking a deep breath, we then find that formal differentiationyields:

dA(t)e(f) = f(t)e(f)dt, (5.1)

and so for suitable operator-valued processes (F(t), t 0) we can define the quan-tum stochastic annihilation integral

t0

F(s)dA(s) by its action on exponentialvectors: t

0

F(s)dA(s)

e(f) =

t0

F(s)f(s)ds

e(f).

The creation integral is obtained by formal adjunction:

e(f),t

0

G(s)dA(s) e(g) = t

0

G(s)dA(s) e(f), e(g)=

t0

f(s)e(f), G(s)e(g),

for each f, g L2(R+). The celebrated quantum Ito formula is summarised in thefollowing table:

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488 DAVID APPLEBAUM

dA(t) dA(t) dtdA(t) dt 0 0

dA(t) 0 0 0dt 0 0 0

These formal relations are suggested by the following type of calculation usingthe CCRs and (5.1):

e(f), dA(t)dA(t)e(g) = dte(f), e(g) + e(f), dA(t)dA(t)e(g)= dte(f), e(g) + dA(t)e(f), dA(t)e(g)= (dt + o(dt))e(f), e(g)

The preceeding heuristic calculations were all given a precise rigorous meaningin the seminal paper [19].

The non-trivial Ito correction term dA(t)dA(t) = dt will only contribute to

formal differentiation of terms that violate Wick ordering. This insight was thebasis of a short note by Hudson and Streater [20] whose title says it all - Itosformula is the chain rule with Wick ordering. They consider processes that takethe Wick-ordered form

M(t) :=j

cjfj(A(t), t)gj(A(t), t),

where each cj C and fj and gj are smooth. Define formal partial derivatives byM(t)

A(t):=j

cj1fj(A(t), t)gj(A(t), t),

M(t)

A(t):=

j

cjfj(A(t), t)1gj(A(t), t)

andM(t)

t:=j

cj [2fj(A(t), t)gj(A(t), t) + fj(A(t), t)2gj(A(t), t)],

where for i = 1, 2, i denotes partial differentiation with respect to the ith variable.The authors then show that

dM(t) =M(t)

A(t)dA(t) +

M(t)

A(t)dA(t) +

M(t)

tdt.

In the two papers [17] and [18], Hudson and Parthasarathy investigate quantumdiffusions. These are prototypes for the quantum stochastic processes (in the senseof [1]) that eventually became known as quantum stochastic flowsor Evans-Hudsonflows (see e.g. [25] and [27]). The authors work in the space h := h0 (L2(R+))where h

0is a complex, separable Hilbert space which carries a representation of

the CCRs. So we have a pair (a, a) of mutually adjoint linear operators acting inh0 and satisfying [a, a

] = 1. This bosonic system is then perturbed by quantumnoise under the constraint that the commutation relation is preserved in time.

So we obtain mutually adjoint processes (a#t , t 0) that satisfy [at, at ] = 1 forall t 0. These are required to be adapted to the Fock filtration in that each

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a#t = a

#1 (t) I where a#1 (t) operates non-trivially on h0 (L2([0, t))). The form

of the perturbation is given by

dat = F(t)dA(t) + G(t)dA(t) + H(t)dt,

and applying quantum Itos formula to the CCRs yields the restraint equations(which are to be read pointwise in t):

[F, a] = [a, G] = 0,

[H, a] + [a, H] = FFGG.Furthermore they obtain formal conditions for the dynamics to be induced by

a unitary operator-valued process (U(t), t 0). Indeed the unitarity requirementimplies the form:

dU(t) = U(t)LdA(t) LdA(t) + iH

1

2LL dt ,

with U(0) = I, where L and H are (ampliations of) linear operators acting on h0with H being formally self-adjoint. In order to obtain

a#(t) = U(t)(a# I)U(t)

for all t 0, it is shown that we must have

F = [L, a], G = [a, L], H = i[H, a] 12

(LLa 2LaL + aLL),

so H = L(a) where L is the Lindblad generator. Indeed it is precisely the generatorof the quantum dynamical semigroup (Tt, t 0) defined by

Tt(X) = E0(U(t)(X

I)U(t)),

for each X B(h0), where E0 denotes the vacuum conditional expectation. Afermionic version of some of these ideas was developed in [4].

As was pointed out above, these ideas were made fully rigorous in [19] whichalso introduced the conservation process7 into quantum stochastic calculus andthus completed the trio of basic integrators. The theory developed therein hasbeen described and extended in a number of monographs and surveys (see e.g.[27], [25], [5], [23]) so the reader will surely forgive me if I stop at this point.

In conclusion, the period 1971-1984 saw a remarkable period of activity fromRobin and his collaborators which led from the quantum central limit theorem toquantum Brownian motion and then to the development of quantum stochasticcalculus. It is perhaps a little unfair to compare this to the gap between thefirst use of the central limit theorem by Abraham de Moivre in 1733 and the

discovery of stochastic calculus by Kiyosi Ito in the 1940s (see [22] for a concisehistorical account of developments leading to the birth of the latter), nonethelessit is certainly a considerable achievement.

7Sometimes called the gauge or number process

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490 DAVID APPLEBAUM

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ON THE CHARACTERISTIC FUNCTION OF RANDOM

VARIABLES ASSOCIATED WITH BOSON LIE ALGEBRAS

LUIGI ACCARDI AND ANDREAS BOUKAS

Abstract. We compute the characteristic function of random variables de-

fined as self-adjoint linear combinations of the Schrodinger algebra generators.We consider the extension of the method to infinite dimensional Lie algebras.

We show that, unlike the second order (Schrodinger) case, in the third order(cube of a Gaussian random variable) case the method leads to a nonlinear

infinite system of ODEs whose form is explicitly determined.

1. Introduction

It is known (see [5] and the bibliography therein) that, while the usual Heisen-berg algebra and sl(2;R) admit separately a continuum limit with respect to theFock representation, the corresponding statement for the Lie algebra generated bythese two, i.e. the Schrodinger algebra is false.

Stated otherwise: there is a standard way to construct a net of Calgebraseach of which, intuitively speaking, is associated to all possible complex valuedstep functions defined in terms of a fixed finite partition of R into disjoint in-tervals (see [6]), but there are obstructions to the natural extension of the Fockrepresentation to this net of Calgebras: these obstructions are called no-go the-

orems. The detailed analysis of these theorems is a deep problem relating thetheory of representations of Lie algebras with the theory of infinitely divisibleprocesses.

To formulate these connections in a mathematically satisfactory way is a prob-lem, which requires the explicit knowledge of all the vacuum characteristic func-tions of the self-adjoint operators associated to the Fock representation of a givensub-algebra of the full oscillator algebra.

For the first order case these characteristic functions have been known for along time and correspond to the standard Gaussian and Poisson distributions onR.

For the full second order case they have been identified in [7] with the threeremaining (i.e. in addition to Gaussian and Poisson) Meixner classes.

Received 2010-9-28; Communicated by D. Applebaum.2000 Mathematics Subject Classification. Primary 60B15, 17B35; Secondary 17B65, 81R05,

81R10.Key words and phrases. Heisenberg algebra, oscillator algebra, sl(2), Schrodinger algebra,

universal enveloping algebra, characteristic function, quantum random variable, Gaussian ran-dom variable, splitting lemma.

493



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494 LUIGI ACCARDI AND ANDREAS BOUKAS

For the Schrodinger algebra they were not known and it is intuitively obviousthat they should form a family of characteristic functions interpolating among allthe five Meixner classes: this is because the generators of the Schrodinger algebraare obtained by taking the union of the generators of the Heisenberg algebra withthose of sl(2;R).

It is also intuitively clear that the class of these characteristic functions cannotbe reduced to those corresponding to the Meixner distributions because these areknown to be infinitely divisible while, from the no-go theorem, one can deduce thatsome of the characteristic functions associated to the Schrodinger algebra cannotbe infinitely divisible.

Our goal is to describe this latter class of characteristic functions. This will pro-vide a deeper insight into the no-go theorems (which up to now have been provedby trial and error, showing that the scalar product, canonically associated to theFock representation, is not positive definite) as well as a deeper understanding ofthe quantum decomposition of some classes of infinitely divisible random variables.

The first step towards achieving this goal is to calculate these characteristicfunctions. This is done by solving some systems of Riccati equations. This leads,in particular, to a representation of the Meixner characteristic functions in a unifiedform that we have not found in the literature (where there are many explicit unifiedforms for some classes of the Meixner distributions). In the last part of the paperwe show that, by applying the same method to hermitian operators involving thecube of creators and annihilators, one obtains an infinite chain of coupled Riccatiequations whose explicit solution at the moment is not known.

2. The Full Oscillator Algebra

Definition 2.1. If a and a are a Boson pair, i.e.

[a, a] := a a a a = 1

then(i) the Heisenberg algebra H is the Lie algebra generated by {a, a, 1}(ii) the sl(2) algebra is the Lie algebra generated by

{a2

, a2, a a +1

2}

(iii) the oscillator algebra O is the Lie algebra generated by

{a, a, a a +1

2, 1}

(iv) the Schrodinger algebra S is the Lie algebra generated by

{a, a

, a

2

, a

2

, a

a +

1

2 , 1}(v) the universal enveloping Heisenberg algebra U (containing H, sl(2), O and Sas sub-algebras) is the Lie algebra generated by

{an

ak ; n, k = 0, 1, 2,...}

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ON THE CHAR. FUNCTION OF RVS ASSOCIATED WITH BOSON LIE ALGEBRAS 495

Defining duality by (a) = a, we can view H, sl(2), O, S and U as -Liealgebras. In particular, H, sl(2), O and S are -Lie sub-algebras of U.

Remark 2.2. The reason for the additive term 12 , in the generators of sl(2), O andS, is explained in the remark following Proposition 2.4.

Lemma 2.3. If a and a are a Boson pair then

[a a, a] = a ; [a, a a] = a

[a2, a2

] = 2 + 4 a a ; [a2, a a] = 2 a2 ; [a a, a2

] = 2 a2

and

[a2, a] = 2 a ; [a, a2

] = 2 a

Proof. This is a well known fact that can be checked without difficulty.

Proposition 2.4. (Commutation relations in S) Using the notation S10 = a,

S01 = a, S20 = a2, S02 = a2, S11 = a a + 12 , S00 = 1 we can write the commutation

relations among the generators of S as

[Snk , SNK] = (k N K n) S

n+N1k+K1 (2.1)

with (Snk )

= Skn, for all n,k,N,K {0, 1, 2} with n + k 2 and N + K 2.

Proof. The proof follows by using Lemma 2.3.

Remark 2.5. The concise formula (2.1) for the commutation relations among thegenerators of S, in particular the inclusion of [S02 , S

20 ] = 4 S

11 , was the reason

behind the inclusion of the additive 12 term in the definition of the generators ofS.

To describe the commutation relations in U, using the notationy,z

x

=y

x

z(x)

,n,k = 1 n,k where n,k is Kroneckers delta, x

(y) = x(x 1) (x y + 1) forx y, x(y) = 0 for x < y, x(0) = 1, we define

L(N, K; n, k) = H(L 1)

K,0 n,0

K, n

L

k,0 N,0

k, N

L

(2.2)

where H(x) is the Heaviside function (i.e., H(x) = 1 if x 0 and H(x) = 0otherwise). Notice that if L exceeds (K n) (k N) then L(N, K; n, k) = 0.

Proposition 2.6. (Commutation relations inU) Using the notationBxy = ax ay,

for all integersn , k , N , K 0

[BNK, Bnk ] =

L1

L(N, K; n, k) BN+nLK+kL

where L(N, K; n, k) is as in (2.2).

Proof. The proof follows from a discretization of Lemma 2.3 of [5] (i.e. by elimi-nating the time indices t and s and replacing the Dirac delta function by 1) andis a consequence of the General Leibniz Rule, Proposition 2.2.2. of [11] .

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Definition 2.7. (The Heisenberg Fock space) The Heisenberg Fock space F isthe Hilbert space completion of the linear span E of the set of exponential vectors

{y() = e a

; C

}, where is the vacuum vector such that a = 0 and|||| = 1, with respect to the inner product

y(), y() = e

H, O, sl(2), S and U can all be represented as operators acting on the exponentialvectors domain E of the Heisenberg Fock space F according to

an

ak y() = kn

n|=0 y( + ) ; n, k = 0, 1, 2,...

If s R and X is a self-adjoint operator on the Heisenberg Fock space Fthen, through Bochners theorem, , ei s X can be viewed as the characteristicfunction (i.e. the Fourier transform of the corresponding probability measure) ofa classical random variable whose quantum version is X.

The differential method (see [11], [10] and Lemma 3.3 below) for the compu-

tation of , ei s X relies on splitting the exponential es X into a product ofexponentials of the Lie algebra generators. In the case of a Fock representation,where the Lie algebra generators are divided into creation (raising) operators, an-nihilation (lowering) operators that kill the vacuum vector , and conservation(number) operators having as an eigenvector, the exponential es X is split intoa product of exponentials of creation operators.

The coefficients of the creation operators appearing in the product exponentialsare, in general, functions of s satisfying certain ordinary differential equations(ODEs) and vanishing at zero. In the case of a finite dimensional Lie algebra (seesection 3) the ODEs can be solved explicitly and the characteristic function of therandom variable in consideration can be explicitly determined.

For infinite dimensional Lie algebras such as, for example, the sub-algebra of

U generated by a

3

and a

3

, the situation is different. As illustrated in section 5,the ODEs form an infinite non-linear system which, in general, is hard (if at allpossible) to solve explicitly.

3. Schrodinger Stochastic Processes

In this section, using the notation of Proposition 2.4, we compute the charac-teristic function , ei s V , s R, of the quantum random variable

V = a S10 + a S01 + b S

20 + b S

02 + S

00 + S

11

where a, b C with b = 0 (the case b = 0 is discussed in the remarks followingProposition 3.4) and , R with 4 |b|2 2 = 0.

The above form of V is the extention of the forms considered in [2]-[4] to thefull Schrodinger algebra. The splitting lemmas for the Heisenberg and sl(2) Lie

algebras can be found in [11]. An analytic proof of the splitting lemma for sl(2)can be found in [8].

Remark 3.1. Here, and in what follows, the terms splitting and disentanglementrefer to the expression of the exponential of a linear combination of operators asa product of exponentials.

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Lemma 3.2. For all a, b C,

(i)

S02 e

a S20

= 4 a2

S20 e

a S20

+ 4 a ea S2

0

S11 + e

a S20

S02

(ii)

S02 eb S1

0 = b2 eb S1

0

(iii)

S01 ea S2

0 = 2 a S10 ea S2

0 + ea S2

0 S01

(iv)

S11 ea S1

0 = a S10 ea S1

0 + ea S1

0 S11

(v)

S11 ea S1

0 =

a S10 +

1

2

ea S

1

0

(vi)

S11 eb S2

0 = 2 b S20 eb S2

0 + eb S2

0 S11

(vii)

S11 eb S2

0 =

2 b S20 +

1

2

eb S

2

0

Proof. The proof of (i) through (iii) can be found in [2]. The proof of (iv) and (vi)follows, respectively, from Propositions 2.4.2 and 3.2.1 of [11] with, in the notationof [11] , D = S01 , X = S

10 , X D = S

11

12 , H = t 1, and f(X) = e

a X for (iv), and

S20 = 2 R, S02 = 2, S

11 = , and f(R) = e

b R for (vi). Finally, (v) and (vii) followfrom the fact that S11 =

12 .

The following lemma is the extention of the splitting lemmas of [1]-[3] to thefull Schrodinger algebra.

Lemma 3.3. For all s C

es V = ew1(s) S2

0 ew2(s) S1

0 ew3(s)

where, letting

K =

4 |b|2 2 ; L = arctan

K

; =

a

2 b

=a

K+ tan L ; = ( cos L + sin L)

we have that

w1(s) =K tan (K s + L)

4 b(3.1)

w2(s) = tan(K s + L) + sec(K s + L) + (3.2)

and

w3(s) = {c0 + c1 s + c2 ln (cos (K s + L))

+c3 ln (sec (K s + L) + tan (K s + L))

+c4 tan(K s + L) + c5 sec(K s + L)} (3.3)

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where

c1 = + a + b (2 2)

c2 =

a + 2 bK

+ 12

c3 =a + 2 b

K

c4 =b (2 + 2)

K

c5 =2 b

Kc0 = (c2 ln (cos L) + c3 ln (sec L + tan L)

+c4 tan L + c5 sec L)

Proof. We will show that w1(s), w2(s) and w3(s) satisfy the differential equations

w1(s) = 4 b w1(s)2 + 2 w1(s) + b

w2(s) = (4 b w1(s) + ) w2(s) + a + 2 a w1(s)

w3(s) = + a w2(s) + 2 b w1(s) + b w2(s)2 +

2

with w1(0) = w2(0) = w3(0) = 0, whose solutions are given by (3.1), (3.2) and(3.3) respectively. We remark that the differential equations defining w1(s) andw2(s) are, respectively, of Riccati and linear type. Moreover, unlike the infinitedimensional case treated in section 5, the ODE for w1 does not involve w2 and w3.

So letE := es (b S

2

0+b S0

2+a S1

0+a S0

1+ S0

0+ S1

1) (3.4)

Then, by the disentanglement assumption,

E = ew1(s) S2

0 ew2(s) S1

0 ew3(s) (3.5)

where wi(0) = 0 for i {1, 2, 3}. Then, (3.5) implies that

sE =

w1(s) S

20 + w

2(s) S

10 + w

3(s)

E (3.6)

and, since by Lemma 3.2,

S02 E = (4 w1(s)2 S20 + 4 w1(s) w2(s) S

10 (3.7)

+2 w1(s) + w2(s)2) E

S01 E =

2 w1(s) S10 + w2(s)

E (3.8)

and

S11 E =

2 w1(s) S

20 + w2(s) S

10 +

1

2

E (3.9)

from (3.4) we also obtain

s E = ((b + 4 b w1(s)

2 + 2 w1(s)) S20 (3.10)

+(a + 2 a w1(s) + 4 b w1(s) w2(s) + w2(s)) S10

+ + a w2(s) + 2 b w1(s) + b w2(s)2 + 2 ) E

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From (3.6) and (3.10), by equating coefficients of S20 , S10 and 1, we obtain (3.1),

(3.2) and (3.3) thus completing the proof.

Proposition 3.4. (Characteristic Function of V) In the notation of Lemma 3.3,for all s R

, ei s V = exp ({c0 + i c1 s + c2 ln (cos (K i s + L))

+c3 ln (sec (K i s + L) + tan (K i s + L))

+c4 tan(K i s + L) + c5 sec(K i s + L)})

Proof. Using the fact that for any z C

ez S0

2 = ez S0

1 =

by Lemma 3.3 we have

, es V = , ew1(s) S2

0 ew2(s) S1

0 ew3(s)

= ew2(s) S0

1 ew1(s) S0

2 , ew3(s)

= , ew3(s)

= ew3(s) ,

= ew3(s)

= exp ({c0 + c1 s + c2 ln (cos (K s + L))

+c3 ln (sec (K s + L) + tan (K s + L))

+c4 tan(K s + L) + c5 sec(K s + L)})

and the formula for the characteristic function of V is obtained by replacing s byi s where s R.

Remark 3.5. In the well known (see [13] and [11]) Gaussian case a = 1, b = = = 0, the differential equations for w1(s), w2(s) and w3(s) in the proof of Lemma

3.3 are greatly simplified and yield

w1(s) = 0 ; w2(s) = s ; w3(s) =s2

2which implies that

, ei s V = , ei s (S1

0+S0

1) = es2

2 (3.11)

i.e. V = S10 + S01 is a Gaussian random variable.

Remark 3.6. In the case when a = 0 we find that = = = 0, c1 = , c2 = 12 ,

c3 = c4 = c5 = 0, c0 =12 ln (cos L) and so

w3(s) =1

2ln (cos L)

1

2ln (cos (K s + L)) + s

Therefore, by Proposition 3.4, assuming that cos L = 0, for all s C

such thatcos(i K s + L) = 0, the characteristic function of V = b S20 + b S02 + S

00 + S

11 is

, ei s V = (cos L)1/2 (sec(i K s + L))1/2 ei s

which, in the case = 0, is the characteristic function of a continuous binomialrandom variable (see [11]).

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Remark 3.7. Ifb = 0 then the ODEs for w1, w2, w3 in the proof of Lemma 3.3 takethe form

w1(s) = 2 w1(s)

w2(s) = w2(s) + a + 2 a w1(s)

w3(s) = + a w2(s) +

2

with wi(0) = 0 for i {1, 2, 3}. Therefore, for = 0

w1(s) = 0

w2(s) =a

(e s 1)

w3(s) =|a|2

2(e s 1) +

+

2+

|a|2

s

Therefore, the characteristic function of V = a S10 + a S01 + S

00 + S

11 is

, ei s V = e|a|

2

2 (ei s1)+i

+ 2

+ |a|2

s

which, for + 2 +|a|2

= 0, reduces to the characteristic function of a Poissonrandom variable (see [13] and also Proposition 5.2.3 of [11]).

4. Random Variables in U: The General Scheme

In analogy with section 3, using the notation Bnk = an ak, we consider quantum

random variables of the form

W =

n,k0

cn,k B

nk + cn,k B

kn

where cn,k C.

Extending the framework and terminology of [11] and [12] to the infinite di-mensional case, the group element (in terms of coordinates of the first kind)

es W = es

n,k0 (cn,k Bnk +cn,k B

kn)

can be put, through an appropriate splitting lemma, in the form of coordinates ofthe second kind

es W =

n,k0

efn,k(s) Bnk efn,k(s) B

kn

for some functions fn,k(s). Since Bnk = 0 for all k = 0, after several commuta-

tions, we find that

es W =n0

ewn(s) Bn0

for some functions wn(s) (we may call them the vacuum coordinates of the secondkind). Therefore, as in Proposition 3.4,

, ei s W = ew0(i s)

For finite-dimensional Lie algebras (see, for example, Lemma 3.3) solving thedifferential equations that define the wns is relatively easy. As shown in the next

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section, in the case of an infinite dimensional Lie algebra the situation is muchmore complex.

5. Example: The Cube of a Gaussian Random Variable

To illustrate the method described in section 4, we will consider the character-istic function , ei s W of the quantum random variable

W = (a + a)3 = (B01 + B10 )

3

By (3.11) W is the cube of a Gaussian random variable. Using a a = 1 + a awe find that

(a + a)2 = a2 + a2

+ 2 a a + 1

and

(a + a)3 = a3 + 3 a + 3 a2

a + 3 a + 3 a a2 + a3

ThusW = a3 + 3 a + 3 a

2a + 3 a + 3 a a2 + a

3

Lemma 5.1. For all analytic functions f and for all wi C, i = 1, 2, 3, 4

[a, f(a)] = f(a) (5.1)

[f(a), a] = f(a) (5.2)

Moreover, if a = 0 then for n 1

an f(a) = f(n)(a) (5.3)

where f(n) denotes the n-th derivative of f.

Proof. The proof of (5.1) and (5.2) can be found in [11]. To prove (5.3) we notice

that, by Proposition 2.6, for n 1 and k 0

[an, ak

] =l1

n

l

k(l) a

klanl

Thus, assuming that f(a) =

k0 ck ak, we have that

an f(a) = [an, f(a)]

=k0

ck [an, a

k]

=k0

ckl1

n

l

k(l) a

klanl

=k0

ck k(n) a

kn

= f(n)(a)

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Corollary 5.2. For an analytic function g and n {1, 2, 3},

an eg(a) = Gn(a

) eg(a)

where

G1(a) = g(a)

G2(a) = g(a) + g(a)

2

G3(a) = g(a) + 3 g(a) g(a) + g(a)3

Proof. The proof follows from Lemma 5.1 by taking f(a) = eg(a).

Corollary 5.3. Let g(a) =

k=0 wk ak. Then, for n {1, 2, 3},

an eg(a) = Gn(a

) eg(a)

where G1(a) =

k=0 (k + 1) wk+1 a

k and

G2(a) =

k=0

(k + 1) (k + 2) wk+2 ak

+

k,m=0

(k + 1) (m + 1) wk+1 wm+1 ak+m

G3(a) =

k=0 (k + 1) (k + 2) (k + 3) wk+3 a

k

+3

k,m=0 (k + 1) (m + 1) (m + 2) wk+1 wm+2 ak+m

+

k,m,=0 (k + 1) (m + 1) ( + 1) wk+1 wm+1 w+1 ak+m+

Proof. The proof follows directly from Corollary 5.2.

Lemma 5.4. Let s C

. Then

es W =

n=0

ewn(s) an

=

n=0

ewn(s) Bn0

where wn(0) = 0 for all n 0 and the wn satisfy the nonlinear infinite system ofdisentanglement ODEs

w0(s) = w1(s)3 + 3 w1(s) (1 + 2 w2(s)) + 6 w3(s)

w1(s) = 3 w1(s)2 (1 + 2 w2(s)) + 18 w3(s) w1(s)

+12 w2(s)2 + 12 w2(s) + 24 w4(s) + 3

w2(s) = 12 w2(s)2 w1(s) + (54 w3(s) + 12 w1(s)) w2(s)

+3 w1(s) + 27 w3(s) + 36 w1(s) w4(s)

+9 w1

(s)2 w3

(s) + 60 w5

(s)

w3(s) = 54 w3(s)2 + (30 w1(s) w2(s) + 18 w1(s)) w3(s)

+24 w4(s) w2(s) + 60 w1(s) w5(s) + 72 w2(s) w4(s)

+12 w1(s)2 w4(s) + 120 w6(s) + 6 w2(s) + 48 w4(s)

+12 w2(s)2 + 1

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and for n 4,

wn(s) = (n + 1) (n + 2) (n + 3) wn+3(s)

+3 (n 1) wn1(s) + 3 (n + 1)2 wn+1(s)

+3

k,m0

k+m=n

(k + 1) (m + 1) (m + 2) wk+1(s) wm+2(s)

+

k,m,0

k+m+=n

(k + 1) (m + 1) ( + 1) wk+1 wm+1 w+1

+3

k,m0

k+m=n1

(k + 1) (m + 1) wk+1(s) wm+1(s)

Proof. As in the proof of Lemma 3.3, let

E := es (a3+3 a+3 a

2a+3 a+3 a a2+a

3) . (5.4)

Then by the disentanglement assumption for es W , we also have that

E =

n=0

ewn(s) an

(5.5)

where wn(0) = 0 for all n 0. Then, (5.5) implies that

sE =

n=0

wn(s) an E (5.6)

and, using Corollary 5.3, from (5.4) we also obtain

sE = (a3 + 3 a + 3 a

2a + 3 a + 3 a a2 + a

3) E

= {k0 (

k+ 1) (

k+ 2) (

k+ 3)

wk+3(

s)

ak

+3

k,m0

(k + 1) (m + 1) (m + 2) wk+1(s) wm+2(s) ak+m

+

k,m,=0

(k + 1) (m + 1) ( + 1) wk+1 wm+1 w+1 ak+m+

+3 a + 3k0

(k + 1) wk+1(s) ak+2

+3k0

(k + 1) wk+1(s) ak + 3

k0

(k + 1) (k + 2) wk+2(s) ak+1

+3

k,m0 (k + 1) (m + 1) wk+1(s) wm+1(s) a

k+m+1

+a3

} E (5.7)

and the differential equations defining the wns are obtained from (5.6) and (5.7)by equating coefficients of the powers of a.

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Remark 5.5. As in the finite dimensional case, w0 is determined by straight-forward integration. However, unlike the finite-dimensional (Schrodinger) caseof Lemma 3.3, in the infinite-dimensional case the disentanglement ODEs are cou-pled, with the ODE for each wn depending on w1,...,wn+3. The ODEs for w1, w2,and w3 are of pseudo (due to coupling)-Riccati type.

Proposition 5.6. (Characteristic function of W) For all s R

, ei s W = ew0(i s)

where w0 is as in Lemma 5.4.

Proof. The proof is similar to that of Proposition 3.4.

References

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2. Accardi, L., Boukas, A.: Random variables and positive definite kernels associated with

the Schroedinger algebra, Proceedings of the VIII International Workshop Lie Theory andits Applications in Physics, Varna, Bulgaria, June 16-21, 2009, pages 126-137, American

Institute of Physics, AIP Conference Proceedings 1243.3. Accardi, L., Boukas, A.: Central extensions and stochastic processes associated with the Lie

algebra of the renormalized higher powers of white noise, with Luigi Accardi, Proceedings

of the 11th workshop: non-commutative harmonic analysis with applications to probability,Bedlewo, Poland, August 2008, Banach Center Publ. 89 (2010), 13-43.

4. Accardi, L., Boukas, A.: The Fock kernel for the Galilei algebra and associated randomvariables, submitted.

5. Accardi, L., Boukas, A., Franz, U.: Renormalized powers of quantum white noise, Infinite

Dimensional Analysis, Quantum Probability, and Related Topics, Vol. 9, No. 1 (2006), 129147.

6. Accardi, L., Dhahri, A.: Polynomial extensions of the Weyl C-algebra submitted to Comm.Math. Phys. , May 2010

7. Accardi, L., Franz, U., Skeide, M.: Renormalized squares of white noise and other non-Gaussian noises as Levy processes on real Lie algebras, Comm. Math. Phys. 228 (2002)

123150 Preprint Volterra, N. 423 (2000)8. Accardi, L., Ouerdiane, H., Rebe, H.: The quadratic Heisenberg group, submitted to Infinite

Dimensional Analysis, Quantum Probability, and Related Topics.

9. Feinsilver, P. J., Kocik, J., Schott, R.: Representations of the Schroedinger algebra andAppell systems, Fortschr. Phys. 52 (2004), no. 4, 343359.

10. Feinsilver, P. J., Kocik, J., Schott, R.: Berezin quantization of the Schrodinger algebra, Infi-

nite Dimensional Analysis, Quantum Probability, and Related Topics, Vol. 6, No. 1 (2003),5771.

11. Feinsilver, P. J., Schott, R.: Algebraic structures and operator calculus. Volumes I and III,Kluwer, 1993.

12. Feinsilver, P. J., Schott, R. Differential relations and recurrence formulas for representationsof Lie groups, Stud. Appl. Math., 96 (1996), no. 4, 387406.

13. Hudson, R. L., Parthasarathy, K. R.: Quantum Itos formula and stochastic evolutions,

Comm. Math. Phys. 93 (1984), 301323.

Luigi Accardi: Centro Vito Volterra, Universita di Roma Tor Vergata, Via diTorvergata, 00133 Roma, Italy

E-mail address: [email protected]

Andreas Boukas: Department of Mathematics, American College o f Greece, Aghia

Paraskevi 15342, Athens, Greece

E-mail address: [email protected]

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QUANTUM FILTERING IN COHERENT STATES

JOHN E. GOUGH AND CLAUS KOSTLER*

Abstract. We derive the form of the Belavkin-Kushner-Stratonovich equa-

tion describing the filtering of a continuous observed quantum system vianon-demolition measurements when the statistics of the input field used forthe indirect measurement are in a general coherent state.

1. Introduction

One of the most remarkable consequences of the Hudson-Parthasarathy quan-tum stochastic calculus [21] is V. P. Belavkins formulation of a quantum theory offiltering based on non-demolition measurements of an output field that has inter-acted with a given system [4, 6, 7, 8]. Specifically, we must measure a particularfeature of the field, for instance a field quadrature, or the count of the field quanta,and this determines a self-commuting, therefore essentially classical, stochasticprocess. The resulting equations have structural similarities with the classicalanalogues appearing in the work of Kallianpur, Striebel, Kusnher, Stratonovich,Zakai, Duncan and Mortensen on nonlinear filtering, see [16, 23, 24, 28]. Thisshowed that the earlier models of repeated quantum photon counting measure-ments developed by Davies [14, 15] could be realized using a concrete theory: thiswas first shown by taking the pure-jump process limit of diffusive quantum filtering

problems [3].There has been recent interest amongst the physics community in quantum

filtering as an applied technique in quantum feedback and control [1, 2, 10, 11, 12,17, 19, 22, 26, 27]. An additional driver is the desire to go beyond the situationof a vacuum field and derive the filter for other physically important states suchas thermal, squeezed, single photon states, etc. In this note we wish to presentthe filter for non-demolition quadrature and photon-counting measurements whenthe choice of state for the input field is a coherent state with intensity function. The resulting filters are a deformation of the vacuum filters and reduce to thelatter when we take 0, this is perhaps to be expected given that the coherentstates have a continuous-in-time tensor product factorization property. We derivethe filters using the reference probability approach, as well as the characteristicfunction approach.

Received 2010-6-14; Communicated by D. Applebaum.2000 Mathematics Subject Classification. Primary 15A15 ; Secondary 15A09, 15A23.

Key words and phrases. Quantum filtering and estimation, coherent state.* This research is supported by the Engineering and Physical Sciences Research Council,

project grant EP/G039275/1.

505



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506 JOHN E. GOUGH AND CLAUS KOSTLER

1.1. Classical Non-linear Filtering. We consider a state based model wherethe state Xt evolves according to a stochastic dynamics and we make noisy obser-vations Y

ton the state. The dynamics-observations equations are the SDEs

dXt = v (Xt) dt + X (Xt) dWproct , (1.1)

dYt = h (Xt) dt + YdWobst (1.2)

and we assume that the process noise Wproc and the observation noise Wobs areuncorrelated multi-dimensional Wiener processes. The generator of the state dif-fusion is then

L = vii + 12

ijXX2ij ,

where XX = XX . The aim of filtering theory to obtain a least squares estimate

for the state dynamics. More specifically, for any suitable function f of the state,we would like to evaluate the conditional expectation

t (f) := E[f (Xt)

| FYt] ],

with FYt] being the -algebra generated by the observations up to time t.1.1.1. Kallianpur-Striebel Formula. By introducing the Kallianpur-Striebel like-lihood function

Lt (x|y) = expt

0

h (xs)

dys 1

2h (xs)

h (xs) ds

for sample state path x = {xs : 0 s t} conditional on a given sample observa-tion y = {ys : 0 s t}, we may represent the conditional expectation as

t (f) =

Cx0 [0,t]

f(xt) Lt (x|y)P[dx]

Cx0 [0,t]

Lt (x|y)P[dx]

y=Y()

=t (f)

t (1)

where P is canonical Wiener measure and

t (f) () =

Cx0 [0,t]

f(xt) Lt (x|Y ())P [dx] .

1.1.2. Duncan-Mortensen-Zakai and Kushner-Stratonovich Equations. Using theIto calculus, we may obtain the Duncan-Mortensen-Zakai equation for the un-normalized filter t (f), and the Kushner-Stratonovich equation for the normalizedversion t (f). These are

dt (f) = t (Lf) dt + t

f h

dYt,

dt (f) = t (Lf) dt +

t

f h t (f) t h dIt,

where (It) are the innovations:

dIt := dYt t (h) dt, I(0) = 0.We note that there exist variants of these equations for more general processesthan diffusions (in particular for point processes which will be of relevance forphoton counting), and for the case where the process and observation noises arecorrelated.

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QUANTUM FILTERING IN COHERENT STATES 507

1.1.3. Pure versus Hybrid Filtering Problems. We remark that we follow the tra-ditional approach of adding direct Wiener noise Wobs to the observations. We couldof course consider a more general relation of the form dY

t= h (X

t) dt +

YdWobs

tbut for constant coefficients Y a simple rearrangement returns us to the abovesetup.

The situation where we envisage dYt = h (Xt) dt + Y (Xt) dWobst , with Y a

known function of the unobserved state, must be considered as being too good tobe true since we can then obtain information about the unobserved state by justexamining the quadratic variation of the observations process, since we then have

dY (t) dY (t)

= Y(Xt)Y (Xt)

dt. For instance, in the case of scalar processes,if we have Y (X) = |X| then knowledge of the quadratic variation yields themagnitude |Xt| of the signal without any need for filtering. Such situations arerarely if ever arise in practice, and one naturally restricts to pure filtering problems.

2. Quantum Filtering

We wish to describe the quantum mechanical analogue of the classical filteringproblem. To begin with, we note that in quantum theory the physical degreesof freedom are modeled as observables, that is self-adjoint operators on a fixedHilbert space h. The observables will generally not commute with each other. Inplace of the classical notion of a state, we will have a normalized vector h andthe averaged of an observable X will be give by the real number |X. (Here wefollowing the physicist convention of taking the inner product | to be linearin the second argument and conjugate linear in the first .) More generally wedefine a quantum state to be a positive, normalized linear functional E on the setof operators. Every such expectation may be written as

E [X] = trh {X}where is a positive trace-class operator normalized so that trh = 1. The operator is referred to as a density matrix. The set of all states is a convex set whoseextreme points correspond to the density matrices that are rank-one projectorsonto the subspace spanned by a unit vectors h.

To make a full analogy with classical theory, we should exploit the mathematicalframework of quantum probability which gives the appropriate generalization ofprobability theory and stochastic processes to the quantum setting. The standardsetting is in terms of a von Neumann algebra of observables over a fixed Hilbertspace, which will generalize the notion of an algebra of bounded random variables,and take the state to be an expectation functional which is continuous in thenormal topology. The latter condition is equivalent to the -finiteness assumptionin probability theory and results in all the states of interest being equivalent to adensity matrix.

In any given experiment, we may only measure commuting observables. Quan-tum estimation theory requires that the only observables that we may estimatebased on a particular experiment are those which commute with the measuredobservables. In practice, we do not measure a quantum system directly, but applyan input field and measure a component of the output field. The input field resultsin a open dynamics for the system while measurement of the output ensures that

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we met so called non-demolition conditions which guarantee that quantum mea-surement process itself does not destroy the statistical features which we wouldlike to infer. We will now describe these elements in more detail below.

2.1. Quantum Estimation. We shall now describe the reference probabilityapproach to quantum filtering. Most of our conventions following the presentationof Bouten and van Handel [12]. For an alternative account, including historicalreferences, see [9].

Let A be a von Neumann algebra and E be a normal state. In a given experimentone may only measure a set of commuting observables {Y : A}. Define themeasurement algebra to be the commutative von Neumann algebra generated bythe chosen observables

M = vN{Y : A} A.We may estimate an observable X A from an experiment with measurement

algebra M if and only if

X M := {A A : [A, Y] = 0, Y M},That is, if it is physically possible to measure X in addition to all the Y. There-fore the algebra vN{X, Y : A} must again be commutative. We may thenset about defining the conditional expectation of estimable observables onto themeasurement algebra.

Definition 2.1. For commutative von Neumann algebra M, the conditional ex-pectationonto M is the map

E[ |M] : M Mdefined by

E[E[X | M] Y] E[XY], Y M. (2.1)

In contrast to the general situation regarding conditional expectations in thenon-commutative setting of von Neumann algebras [25], this particular definition isalways nontrivial insofar as existence is guaranteed. Introducing the norm A2 :=E[AA], we see that the conditional expectation always exists and is unique up tonorm-zero terms. It moreover satisfies the least squares property

X E[X | M] X Y, Y M.As the set vN{X, Y : A} is a commutative von Neumann algebra for eachX M, it will be isomorphic to the space of bounded functions on a measurablespace by Gelfands theorem. The state induces a probability measure on this spaceand we may obtain the standard conditional expectation of the random variablecorresponding to X onto the -algebra generated by the functions corresponding

to theY. This classical conditional expectation then corresponds to a uniqueelement E[X | M] M and this gives the construction of the quantum conditional

expectation. We sketch the conditional expectation in figure 1. Note that whilethis may seem trivial at first sight, it should be stressed that the commutant M

itself will typically be a non-commutative algebra, so that while our measuredobservables commute, and what we wish to estimate must commute with our

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measured observables, the object we can estimate need not commute amongstthemselves.

'&

$%

?

rrXE[X|M]M

M

Figure 1. Quantum Conditional Expectation E[|M] as a leastsquares projection onto M from its commutant.

The following two lemmas will be used extensively, see [12] and [13].

Lemma 2.2 (Unitary rotations). LetU be unitary and defineE [X] := E UXUand letM = UMU. Then

E[UXU | M] = UE [X | M] U.Here we think of going from the Schrodinger picture where the state E is fixed

and observables evolve to UXU, to the Heisenberg picture where the state evolvesto E and the observables are fixed. Lemma 2.2 tells us how we may transform theconditional expectation between these two picture.

Lemma 2.3 (Quantum Bayes formula). Let F M withE FF = 1 and setEF [X] := E

FXF

. Then

EF [X|M] =

E

FXF | M

E [FF | M] .Proof. For all Y M,

EE

FXF | MY = E FXF Y= EF[XY ], since[F, Y] = 0,

= EF [EF [X | M] Y]= E

FFEF [X | M] Y

, since F M

= EE

FF | MEF [X | M] Y .

Note that the proof only works if F M!2.2. Quantum Stochastic Processes.

We begin by reviewing the theory ofquantum stochastic calculus developed by Hudson and Parthasarathy [21] whichgives the mathematical framework with which to generalize the notions of theclassical Ito integration theory.

We take R+ = [0, ). We shall denote by L2symm(Rn+) the space of all square-integrable functions of n positive variables that are completely symmetric: that

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is, invariant under interchange of any pair of its arguments. The Bose Fock spaceover L2(R+) is then the infinite direct sum Hilbert space

F :=n=0

L2symm(Rn+)

with the n = 0 space identified with C. An element ofF is then a sequence = (n)

n=0 with n L2symm(Rn+) and 2 =

n=0

[0,)n

|n (t1, , tn) |2dt1 dtn < . Moreover, the Fock space has inner product

| =n=0

[0,)n

n (t1, , tn) (t1, , tn) .

Th physical interpretation is that = (n)n=0 Fdescribes the state of a quan-

tum field consisting of an indefinite number of indistinguishable (Boson) particleson the half-line R+. A simple example is the vacuum vector defined by

:= (1, 0, 0,

)

clearly corresponding to no particles. (Note that the no-particle state is a genuinephysical state of the field and is not just the zero vector ofF!) An important classof vectors are the coherent states () defined by

[ ()]n (t1, , tn) := e2 1

n!(t1) (tn),

for L2(R+). (The n = 0 component understood as e2 .) The vacuum thencorresponds to (0).

For each t > 0 we define the operators of annihilation B (t), creation B (t) andgauge (t) by

[B (t) ]n (t1, , tn) :=

n + 1 t

0

n+1 (s, t1, , tn) ds,

[B (t) ]n (t1, , tn) :=1n

nj=1

1[0,t](tj)n1

t1, , tj , , tn ,[ (t) ]n (t1, , tn) :=

nj=1

1[0,t](tj)n (t1, , tn) .

The creation and annihilation process are adjoint to each other and the gauge isself-adjoint. We may define a field quadrature by

Q (t) = B (t) + B (t)

and this yields a quantum stochastic process which is essentially classical in thesense that it is self-adjoint and self-commuting, that is [Q (t) , Q (s)] = 0 f orall t, s

0. We remark that for each

[0, 2) we may define quadratures

Q (t) = eiB (t) + eiB (t) which again yield essentially classical processes,however different quadratures will not commute! For the choice of the vacuumstate, {Q (t) : t 0} then yields a representation of the Wiener process: for realk ()

| ei

0k(t)dQ(t) = e 12

0k(t)2dt.

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Table 1. Quantum Ito Table

dB d dB dt

dB 0 dB dt 0d 0 d dB 0

dB 0 0 0 0

dt 0 0 0 0

.

We also note that { (t) : t 0} is also an essentially classical process and forthe choice of a coherent state yields a non-homogeneous Poisson process: for realk ()

() | ei

0k(t)d(t) () = exp

0

|(t) |2

eik(t) 1

dt.

We consider a quantum mechanical system with Hilbert space h being driven byan external quantum field input. the quantum field will be modeled as an idealized

Bose field with Hilbert space L2 (R+, dt) which is the Fock space over the one-particle space L2 (R+, dt). Elements of the Fock space may be thought of as vectors = n=0n where n = n (t1, , tn) is a completely symmetric functions with

n=0

[0,)n

|n (t1, , tn) |2dt1 dtn < .

The Hudson-Parthasarathy theory of quantum stochastic calculus gives a gen-eralization of the Ito theory of integration to construct integral processes withrespect to the processes of annihilation, creation, gauge and, of course, time. Thisleads to the quantum Ito table 1.

We remark that the Fock space carries a natural filtration in time obtainedfrom the decomposition F= Ft] F(t into past and future subspaces: these arethe Fock spaces over L2 [0, t] and L2(t,

) respectively.

2.3. Continuous-Time Quantum Stochastic Evolutions. On the joint spaceh F, we consider the quantum stochastic process V() satisfying the QSDE

dV(t) =

(S I) d(t) + L dB (t)

LS dB (t) ( 12 LL + iH) dt

V(t),

with V(0) = 1, and where S is unitary, L is bounded and H self-adjoint. Thisspecific form of QSDE may be termed the Hudson-Parthasarathy equationas thealgebraic conditions on the coefficients are necessary and sufficient to ensure uni-tarity (though the restriction for L to be bounded can be lifted). The processis also adapted in the sense that for each t > 0, V(t) acts non-trivially on thecomponent h Ft] and trivially on F(t.2.3.1. The Heisenberg-Langevin Equations. For a fixed system operator X we set

jt (X) := V (t) [X I] V(t) . (2.2)

Then from the quantum Ito calculus we get

djt (X) = jt (L11X) d (t) +jt (L10X) dB (t))+jt (L01X) dB (t) +jt (L00X) dt (2.3)

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d dinput

systemoutput

Figure 2. Input - Output component

where the Evans-Hudson maps L are explicitly given byL11X = SXS X,L10X = S[X, L],L01X = [L, X]SL00X = L(L,H)

and in particular

L00 takes the generic form of a Lindblad generator:

L(L,H) = 12

L[X, L] + 12

[L, X]L i [X, H] . (2.4)2.3.2. Output Processes. We introduce the processes

Bout (t) := V (t) [I B (t)] V(t) ,out (t) := V (t) [I (t)] V(t) . (2.5)

We note that we equivalently have Bout (t) V (T) [1 B (t)] V(T), for t T.Again using the quantum Ito rules, we see that

dBout = jt(S)dB(t) +jt(L)dt,

dout = d (t) +jt(LS)dB(t) +jt(S

L)dB(t) +jt(LL)dt. (2.6)

2.3.3. The Measurement Algebra. We wish to consider the problem of continu-ously measuring a quantum stochastic process associated with the output field.we shall chose to measure an observable process of the form

Yout(t) := V(t)[I Yin(t)]V(t) (2.7)which corresponds to a quadrature of the field when

Yin(t) = Q (t) = B(t) + B(t),

or counting the number of output photons when

Yin(t) = (t).

We introduce von Neumann algebra

Yint] = vN

Yin (s) : 0 s t

,

and define the measurement algebra up to time t to beYoutt] = vN

Yout (s) : 0 s t V(t)Yint] V(t) . (2.8)

Note that both algebras are commutative:Yout (t) , Yout (s)

= V(T)

I Yin (t) , Yin (s) V(T) = 0

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for T = ts. The familyYoutt] : t 0

then forms an increasing family (filtration)

of von Neumann algebras.

2.3.4. The Non-Demolition Property. The system observables may be estimatedfrom the current measurement algebra

jt (X) Youtt]

. (2.9)

The proof follows from the observation that for t sjt (X) , Y

out (s)

= V(t)

X I, I Yin (s)V(t) = 0.2.4. Constructing The Quantum Filter. The filtered estimate forjt (X) giventhe measurements of the output field is then

t (X) := E jt (X) | Youtt] .Let Et [X] = E [jt (X)], then by lemma 2.2

t (X) = Ejt (X) | Youtt]

= V(t)

Et

X|Yint]

V(t) .

2.4.1. Reference Probability Approach. Suppose that there is an adapted process

F () such that F (t) Yint]

and Et [X|] = E

F (t)

(X 1) F (t)

for all system

operators X, then by lemma 2.3

t (X) = E jt (X) | Youtt] = V(t)

Et

X | Yint]

V(t)

= V(t)E

F (t)

(X 1) F (t) | Yin

t]

E

F (t)

F (t) | Yin

t]

V(t)This is essentially a non-commutative version of the Girsanov transformation fromstochastic analysis. The essential feature is that the transformation operators F (t)giving the change of representation for the expectation lie in the commutant ofthe measurement algebra up to time t.

We therefore obtain an operator-valued Kallianpur-Striebel relation

t (X) =t (X)

t (1) , (2.10)

which may be called the quantum Kallianpur-Striebel, where

t (X) := V(t)E

F (t)

(X 1) F (t) | Yint]

V(t) . (2.11)

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3. Coherent State Filters

We shall consider the class of states

E [ ] = | (3.1)

of the form

= () (3.2)where is a normalized vector in the system Hilbert space and () is the coherentstate with test function L2[0, ). We note that

dB (t) () = (t) dt () ,

d (t) () = (t) dB(t) () .

We see that

E [djt (X)] = E

[jt L(t)X]dt (3.3)where

L(t)X = L00X + (t) L10X + (t) L01X + |(t) |2L11X (3.4)The generator is again of Lindblad form and in particular we have

L(t) L(L(t),H)with

L(t) = S(t) + L. (3.5)

e may define a parameterized family of density matrices on the system by settingtrh {tX} = E[jt (X)], in which case we deduce the master equation

t =

L(t) () ,

where the adjoint is defined through the duality trh { LX} =trh {L X}.From the input-output relation for the field

dBout = jt (S) dB (t) +jt (L) dt

we obtain the average

E [dBout] = {jt (S) (t) +jt (L)} dt

= jt

L(t)

dt.

3.1. Quadrature Measurement. We take Yin (t) = B (t) + B (t) which is aquadrature of the input field. Setting (t) = V(t) we have that

d (t) = ((S 1) (t) + L) dB (t) (t) (LS(t) + 12

LL + iH)dt (t) (3.6)

At this stage we apply a trick which is essentially a quantum Girsanov transfor-mation. This trick is due to Belavkin [5] and Holevo [20]. We now add a termproportional to dB (t) (t) to get

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d (t) = ((S

1) (t) + L) [dB (t) + dB (t)] (t)

LS(t) + 12

LL + iH + ((S 1) (t) + L) (t) dt (t) LtdY in (t) (t) + Ktdt (t) ,

where

Lt = L + (S I) (t) = L(t) (t) ,Kt = LS(t) 1

2LL iH L(t)(t) + (t)2 .

It follows that (t) F (t) where F (t) is the adapted process satisfying theQSDE

dF (t) = LtdYin (t) F (t) + Ktdt F (t) , F (0) = I.

Moreover F (t) is in the commutant ofYint] and therefore allows us to perform the

non-commutative Girsanov trick.From the quantum Ito product rule we then see that

d [F (t) XF (t)] = F (t)

XLt + LtX

F (t) dY in (t)

+F (t)

LtXLt + XKt + KtX

F (t) dt

and this leads to the SDE for the un-normalized filter

dt (X) = t

XLt + L

tX

dY out (t)

+t

LtXLt + XKt + KtX

dt.

After a small bit of algebra, this may be written in the formdt (X) = t

XLt + L

tX

dY out (t) (t) + (t) dt+ t L(t)X dt.

This is the quantum Zakai equation for the filter based on continuous measurementof the output field quadrature.

To obtain the quantum Kushner-Stratonovich equation we first observe thatthe normalization satisfies the SDE

dt (I) = t

Lt + L

t

dY out (t) (t) + (t) dt

and by Itos formula

d1

t (I)

=

t

Lt + L

t

t (I)2 dY out (t) (t) + (t) dt+t

Lt + L

t

2

t (I)3 dt.

The product rule then allows us to determine the SDE for t (X) =t (X)

t (I):

dt (X) = t

L(t)X

+ dt

t

XLt + L

tX

t (X) t

Lt + L

t

dI(t) ,

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where the innovations process satisfies

dI(t) = dY out (t)

t Lt + Lt + (t) + (t) dt.

We note that the innovations is martingale with respect to filtration generatedby the output process for the choice of probability measure determined by thecoherent state.

3.1.1. The Quadrature Measurement Filter for a Coherent state. In it convenientto write the filtering equations in terms of the operators L(t). The result is theBelavkin-Kushner-Stratonvich equationfor the filtered estimate based on optimalestimation of continuous non-demolition field-quadrature measurements in a co-herent state ().dt (X) = t

L(t)X

dt

+t XL(t) + L(t)X t (X) t L(t) + L(t) dIquad (t) ,(3.7)

with the innovations

dIquad (t) = dY out (t) t

L(t) + L(t)

dt. (3.8)

3.2. Photon Counting Measurement. For convenience we shall derive thefilter based on measuring the number of output photons under the assumptionthat the function is bounded away from zero. We discuss how this restrictioncan be removed later. We now set Yin (t) = (t). We again seek to construct anadapted process F (t) in the commutant ofYint] such that (t) = F (t) . We start

with 3.6 again but now note that

dB (t) (t) = 1(t)

d(t) (t)

and making this substitution gives

d (t) =1

(t)LtdY

in(t) (t)

1

2LL + iH + LS(t)

dt (t) .

We are then lead to the Zakai equation

dt (X) =1

|(t) |2 t

LtXLt + (t)XLt + L

tX(t)

dY out (t)

t

1

2XLL +

1

2LLX + i [X, H] XLS(t) (t) SLX

dt

which may be rearranged as

dt (X) = tLX dt

+1

|(t) |2 t

LtXLt + (t)XLt + L

tX (t)

dY out (t) |(t) |2dt .

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To determine the normalized filter, we note that

dt (I) =1

|(t) |2t Lt Lt + (t)Lt + Lt(t) dY out (t) |(t) |2dt

and that

d1

t (I)=

t

Lt Lt + (t)

Lt + Lt(t)

t (I)

2 dt

t

Lt Lt + (t)

Lt + Lt(t)

t (I)

t

Lt Lt + (t)

Lt + Lt(t)

+ |(t) |2

dY out (t) .Applying the Ito product formula yields the quantum analogue of the Kushner-Stratonovich equation for the normalized filter;

dt (X) = t LXdt + 1t Lt Lt + (t)Lt + Lt(t) + |(t) |2 dI(t)

t(LtXLt + (t)

XLt + LtX(t)) t (X) t(Lt Lt + (t)Lt + Lt(t))

and the innovations process is now

dI(t) = dY out (t)

t

Lt Lt + (t)

Lt + Lt(t)

+ |(t) |2

dt.

We note that the innovations is again a martingale with respect to filtration gen-erated by the output process for the choice of probability measure determined bythe coherent state.

The derivation above relied on the assumption that (t) = 0, however this isnot actually essential. In the case of a vacuum input, it is possible to apply anadditional rotation W (t) satisfying dW (t) = [zdB (t)

zdB (t)

12

|z

|2dt]W (t) ,

with W (0) = I, and apply the reference probability technique to the von Neumannalgebra generated by N(t) = W (t) (t) W (t)

: this leads to a Zakai equation

that explicitly depends on the choice of z C however the Kushner-Stratonovichequation for the normalized filter will be z-independent. Similarly for the generalcoherent state considered here, we could take z to be a function of t in which casewe must chose z so that (t) + z (t) = 0. The Kushner-Stratonovich equationobtained will then be identical to what we have just derived.

3.2.1. Photon Counting Measurement in a Coherent State. Again it is convenientto re-express the filter in term of L(t). We now obtain the Belavkin-Kushner-Stratonvich equation for the filtered estimate based on optimal estimation of con-tinuous non-demolition field-quanta number measurements in a coherent state ().

dt (X) = t L(t)X dt +t L(t)XL(t)t

L(t)L(t) t (X) dInum (t) , (3.9)

with the innovations

dInum (t) = dY out (t) t

L(t)L(t)

dt. (3.10)

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4. Characteristic Function Approach

As an alternative to the reference probability approach, we apply a method

based on introducing a process C(t) satisfying the QSDE

dC(t) = f(t) C(t) dY (t) , (4.1)

with initial condition C(0) = I. Here we assume that f is integrable, but otherwisearbitrary. This approach is a straightforward extension of a classical procedureand as far as we are aware was first used in the quantum domain by Belavkin [4].The technique is to make an ansatz of the form

dt (X) = Ft (X) dt + Ht (X) dY (t) (4.2)where we assume that the processes Ft (X) and Ht (X) are adapted and lie in Yt].These coefficients may be deduced from the identity

E [(t (X) jt (X)) C(t)] = 0which is valid since C(t) Yt]. We note that the Ito product rule implies I +II + III = 0 where

I = E [(dt (X) djt (X)) C(t)] ,II = E [(t (X) jt (X)) dC(t)] ,

III = E [(dt (X) djt (X)) dC(t)] .We illustrate how this works in the case of quadrature and photon counting in

a coherent state. For convenience of notation we shall write St for jt (S), etc.

4.1. Quadrature Measurement. Here we have

dY (t) = StdB (t) + St dB (t)

+ (Lt + L

t ) dt

so that

I = E [Ft (X) C(t) + Ht (X) (Stt + St t + Lt + Lt ) C(t)] dtE

(L00X)t + (L01X)t t + (L10X)t t + (L11X)t |t|2

C(t)

dt,

II = E [(t (X) Xt) f(t) C(t) (Stt + St t + Lt + Lt )] dt,III = E [{Ht (X) (L01X)t St t (L11X)t St t } f (t) C(t)] dt.

Now from the identity I+ II+ III = 0 we may extract separately the coefficients

of f(t) C(t) and C(t) as f(t) was arbitrary to deduce

t ((t (X) Xt) (Stt + St t + Lt + Lt ))+t (Ht (X) (L01X)t St t (L11X)t St t ) = 0,

t

Ft (X) + Ht (X) (Stt + St t + Lt + Lt )

L(t)X

t

= 0.

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Using the projective property of the conditional expectation (t t = t) and theassumption that Ft (X) and Ht (X) lie in Yt], we find after a little algebra that

Ht (X) = t XL(t) + L(t)X t (X) t L(t) + L(t) ,Ft (X) = t

L(t)X

Ht (X) t

L(t) + L(t)

,

so that the equation (4.2) reads as

dt (X) = t

L(t)X

dt + Ht (X)

dY (t) t

L(t) + L(t)

dt

.

4.2. Photon Counting Measurement. We now have

dY (t) = d (t) + LtStdB (t) + St LtdB (t)

+ LtLtdt

so that

I = EFt (X) + Ht (X)|t|2 + LtStt + St Ltt + LtLtC(t) dt

E

L(t)X

t

C(t)

dt

II = E

(t (X) Xt) f(t) C(t)

|t|2 + LtStt + St Ltt + LtLt

dt,

III = E

Ht (X)

|t|2 + LtStt + St Ltt + LtLt

f (t) C(t)

dt

E [{(L01X)t St Lt + (L01X)t t} f(t) C(t)] dtE

(L11X)t |t|2 + (L11X)t St Ltt

f(t) C(t)

dt.

This time, the identity I + II + III = 0 implies

t (t (X) Xt) L(t)t L(t)t + Ht (X) L(t)t L(t)t t

(L01X)t St L(t)t + (L11X)t St L(t)t t

= 0,

t

Ft (X) + Ht (X) L(t)t L(t)t

L(t)X

t

= 0.

Again, after a little algebra, we find that

Ht (X) =t

L(t)XL(t)

t

L(t)L(t) t (X) ,

Ft (X) = tL(t)X

Ht (X) t

L(t)L(t)

,

so that the equation (4.2) reads as

dt (X) = t

L(t)X

dt + Ht (X)

dY (t) t

L(t)L(t)

dt

.

In both cases, the form of the filter is identical to what we found using thereference probability approach.

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5. Conclusion

Both the quadrature filter (3.7) and the photon counting filter (3.9) take on the

same form as in to the vacuum case and of course reduce to these filters when weset 0. In both cases it is clear that averaging over the output gives

E [dt (X)] = E

jt

L(t)X

dt

which is clearly the correct unconditioned dynamics in agreement with (3.3), andwe obtain the correct master equation.

It is worth commenting on the fact that the pair of equations now replacingthe dynamical and observation relations (1.1,1.2) are the Heisenberg-Langevinequation (2.3) and the appropriate component of the input-output relation (2.6).The process and observation noise have the same origin however the nature ofthe quantum filtering based on a non-demolition measurement scheme results ina set of equations that resemble the uncorrelated classical Kushner-Stratonovichequations.

5.1. Is Quantum Filtering still a Pure Filtering Problem? The form of theinput-output relations (2.6) might suggest that it is possible to learn somethingabout the system dynamics by examining the quadratic variation of the outputprocess, however, this is not the case! We in fact have an enforced too good to betrue situation here as the output fields satisfy the same canonical commutationrelations as the inputs with the result that the quantum It o table for the outputprocesses Bout, B

out and out has precisely the same structure as table 1. Therefore

we always deal with a pure filtering theory in the quantum models considered here.

Acknowledgment. The authors gratefully acknowledge the support of the UKEngineering and Physical Sciences Research Council under grant EP/G039275/1.

References

1. Armen, M. A., Au, J. K, Stockton, J. K., Doherty A. C., and Mabuchi, H.: Adaptive

homodyne measurement of optical phase. Phys. Rev. Lett., 89:133602, (2002).2. Barchielli, A.: Direct and heterodyne detection and other applications of quantum stochastic

calculus to quantum optics. Quantum Opt., 2 (1990) 423441.

3. Barchielli, A., Belavkin, V. P.: Measurements continuous in time and posterior states inquantum mechanics, J. Phys. A, Math Gen, 24 (12), 14951514 (1991)

4. Belavkin, V. P.: Quantum filtering of Markov signals with white quantum noise. Radiotech-

nika i Electronika, 25 (1980

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