Igor G.VLADIMIROV

College of Engineering and Computer Science,Australian National University,Canberra,ACT 2601,Australia

A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems

Igor G.VLADIMIROV

College of Engineering and Computer Science,Australian National University,Canberra,ACT 2601,Australia

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements.The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state.The Hamiltonian and system-field coupling operators,as functions of the system variables,are assumed to be represented in a Weyl quantization form.Using the Wigner-Moyal phase-space framework,we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function(QCF)of the system conditioned on the measurements.This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements.We discuss a specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.

Quantum stochastic system,quantum filtering equation,Gaussian approximation

1 Introduction

Estimation of the unknown current state of a stochastic system,based on the past history of a statistically dependent random process,is the central problem in the stochastic filtering theory which dates back to the worksofKolmogorov and Wienerofthe 1940s[1,2].The performance of state estimators is usually quantified by mean square values of the estimation errors which have to be minimized.In the framework of quadratic cost functionals,optimal estimators are delivered by conditional expectations of the state of the system,condi-tioned on the available observation history[3].For linear system-observation dynamics driven by white noise processes[4,5],the mean square optimal estimators found a recursive implementation in the Kalman filter in the 1960s,and their analogues were subsequently developed for robust filtering problems.The latter include,for example,the H∞-settings[6],which employ operator norms with respect to square summable disturbances,and deterministic and statistical uncertainty descriptions based on integral quadratic constraints[7]and entropy theoretic constructs[8,9].

The filtering problems arise naturally when the current system state(whose knowledge,precise or approximate,is required for feedback control)is not accessible to direct measurement.Such measurements are particularly problematic in regard to physical systems on the atomic scales,whose evolution is described in terms of operator-valued variables and obeys the laws of quantum mechanics which prohibit simultaneous measurements of noncommuting quantities[10-12].The incompatibility of quantum variables restrict information on the system which can be retrieved without disturbing it.This reflects the invasive nature of measurement as an interaction with a macroscopic apparatus,which affects the quantum system and is accompanied with conversion of operator-valued processes to real-valued signals.The issue of the quantum information loss is one of the motivations for coherent quantum control[13,14]by measurement-free interconnection,where controllers and observers for quantum plants are also quantum systems.This approach is an active area of research in the quantum linear systems theory[15]and is concerned,in particular,with coherent quantum counterparts[16-23]of the classical H∞and LQG controllers and observers.

Nevertheless,for a class of open quantum systems which are weakly coupled to external electromagnetic fields,the measurement of the output fields can be arranged in a nondemolition manner[24],so that,at any moment of time,the past observations commute between themselves and with future system operators.In this case,the quantum measurements are,in many respects,similar to classical observations and,in fact,can be regarded as classical random processes[25]on a common probability space.The statistical dependence on the system variables,which results from the systemfield interaction,allows such observations to be used for continuously updating the conditional density operator of the quantum system according to the stochastic master equation(SME)[26-28].This posterior density operator(and its modifications)plays the role of an information state in measurement-based quantum control and filtering problems[29-34]involving also quadratic-exponential performance criteria.The SME is a quantum analogue of the Kushner-Stratonovich equation[3]for the evolution of the posterior probability density function(PDF)of the system variables in the case ofclassicalsystem-observation dynamicsdescribed by stochastic differential equations(SDEs).Similarly to its classical counterpart,the SME is a recursive implementation of the Bayesian inference approach.Accordingly,the quantum Belavkin-Kushner-Stratonovich equation(BKSE)[35-38],which governs the dynamics of the conditional expectations of system operators(and is,therefore,dualto the SME),isdeveloped in the framework of the Hudson-Parthasarathy calculus of quantum stochastic differential equations(QSDEs)[39,40].

In the QSDE model of open quantum systems,the external input bosonic fields are represented by timevarying operatorsacting on a symmetric Fock space[41].Furthermore,this Hilbert space is endowed with a quantum state which determines the statistical properties of the fields.The resulting quantum Wiener processes on the Fock space drive the system variables according to the energetics of the system and its interaction with the fields.The latter is specified by the system Hamiltonian and the system-field coupling operators which are functions of the system variables.However,classical functions of several real or complex variables can be extended to the noncommutative quantum variables in different ways.One of such extensions is provided by the Weylfunctional calculus[42]which employs unitary Weyl operators whose role in this context is similar to that of the spatial harmonics in the Fourier transform.

The Weyl quantization is used in the Wigner-Moyal phase-space method[43,44]of quasi-probability density functions(QPDFs)which are the Fourier transforms of the quasi-characteristic functions(QCFs)[45],with the latter being the quantum expectations of the Weyl operators.The phase-space approach allows the quantum dynamics to be represented without the“burden of the Hilbert space”and leads to partial differential and integro-differential equations for the QPDFs and QCFs,which are real or complex-valued functions of several real variables encoding the moments of the system operators.Although the Moyal equations[44]for the QPDF dynamics were originally obtained for closed systems,the phase-space approach has also extensions to different classes of open quantum systems;see,for example,[46-50].

In the present paper,the phase-space approach is applied to the filtering problem for a class of nonlinear quantum stochastic systems with multichannel field measurements satisfying the nondemolition conditions.The system variables satisfy the Weylcanonicalcommutation relations and are governed by a Markovian QSDE driven by the quantum Wiener processes of bosonic fields in vacuum state.Using the Weyl quantization of the Hamiltonian and system-field coupling operators in combination with the results of[38]and[50],we obtain a stochastic integro-differential equation for the evolution of the posterior QCF of the system conditioned on the measurements.This equation is a spatial frequency domain representation of the BKSE driven by the innovation process associated with the measurements.We also discuss a more specific form of the posterior QCF and QPDF dynamics for a class[50,51]of open quantum systems whose coupling operators are linear functions of the system variables while the Hamiltonian is split into a quadratic part and a nonquadratic part represented in the Weyl quantization form.For this linear system-field coupling case,we outline modified quantum Kalman filter equations for a Gaussian approximation ofthe posterior system state.The Weylquantization ofthe Hamiltonian and coupling operators has also been used in[52,53]in a different context of optimality conditions[54,55]for the coherent quantum control and filtering problems mentioned above.

The paper is organized as follows.Section 2 describes the class of quantum stochastic systems under consideration.Section 3 specifies the model of nondemolition measurements and describes the BKSE for conditional expectations.Section 4 applies this equation to the Weyl operators and obtains the posterior QCF dynamics in the Weyl quantization framework.Section 5 specifies these results,together with a related equation for the posterior QPDF,for the case of linear systemfield coupling.Section 6 develops modified quantum Kalman filter equations for a Gaussian approximation of the posterior quantum state.Section 7 provides concluding remarks.

2 Quantum stochastic systems being considered

We consider an open quantum system,whose internal dynamics are affected by interaction with external fields and are described in terms of an even numbernof dynamic variablesX1,...,Xnassembled into a vectorX:=(Xk)1?k?n(vectors are organized as columns).These system variables are time-varying self-adjoint operators on a complex separable Hilbert space H satisfying the canonical commutation relations(CCRs)

which are closely related to the Baker-Campbell-Hausdorffformula foroperator exponentials(see,forexample,[27,pp.128-129])and are represented in terms of the following unitary Weyl operators[42]:

where(·)?denotes the operator adjoint.Here,Θ is a constant nonsingular real antisymmetric matrix of ordernwhich specifies the commutator matrix

as an infinitesimal form of the Weyl CCRs(1)(the transpose(·)Tacts on matrices of operators as if their entries were scalars).

then the CCR matrix takes the form

and corresponds to the symplectic structure matrix in classical Hamiltonian systems(here,?is the Kronecker product of matrices,andIrdenotes the identity matrix of orderr).

The evolution of the vectorXof system variables is governed by a Markovian Hudson-Parthasarathy QSDE with the identity scattering matrix[39,40]

whose structure is described below(the time arguments are omitted for brevity).Although it resembles classical SDEs[25],the QSDE(4)is driven by a vectorW:=(Wk)1?k?mof an even numbermof self-adjoint quantum Wiener processesW1,...,Wmacting on a symmetric Fock space F.These represent the external bosonic fields[40,56]and satisfy the quantum It?o relations

In contrastto the identity diffusion matrix ofthe standard Wiener process,Ω :=(ωjk)1?j,k?mis a complex positive semi-definite Hermitian matrix with an orthogonal antisymmetric imaginary part ImΩ=J(so thatJ2=-Im),and hence,the quantum Wiener processesW1,...,Wmdo not commute with each other:

Furthermore,then-dimensional drift vector L(X)and the dispersion(n×m)-matrix-i[X,hT]of the QSDE(4)are specified by the system Hamiltonianh0and the vectorh:=(hk)1?k?mof system-field coupling operatorsh1,...,hm,which are self-adjoint operators on H representable as functions of the system variablesX1,...,Xn.The superoperator L in(4),which is usually referred to as the Gorini-Kossakowski-Sudarshan-Lindblad(GKSL)generator[57,58],is a quantum analogue of the infinitesimal generators of classical Markov diffusion processes[59].This superoperator acts on a system operator ξ as

and applies to vectors of operators entrywise.The specific structure of the QSDE(4)comes from the system-field interaction which drives a unitary operatorU(t)acting on the system-field tensor-product space H:=H0?F(with H0the initial space for the action of the system variables at timet=0):

The QSDE(4)can be obtained from(9)by using(8)and the quantum It?o formula[39,40]in combination with(5)and commutativity between the forward It?o increments dW(t)and adapted processes taken at times?t.Adapted processes ξ,which are functions of the system variables,satisfy QSDEs of the same form

The special structure of the drift and diffusion terms of these QSDEs gives rise to physical realizability conditions for linear quantum stochastic systems in the state-space[16]and frequency[60]domains and for nonlinear quantum systems[61].Endowed with additional features(including more general scattering matrices with photon exchange between the fields),such QSDEs are employed in a unified formalism for modelling interconnections of quantum systems which interact with each other and the environment[62].Furthermore,the specific structure of the QSDEs plays an important role for dissipativity and other properties of such systems[63,64].

3 Output fields,nondemolition measurements and conditioning

which is part of the diffusion term in(8),is related to a different vectorL:=(Lk)1?k?m/2of(not necessarily self-adjoint)coupling operatorsL1,...,Lm/2by

Here,use is made of the propertyJ2=-Imof the matrixJin(5).The relations(11)and(12)can be used in order to move between two alternative representations of the external fields and the system-field coupling operators.As a result of the joint system-field evolution described by the unitary operatorU(t)from(8),the output fieldY:=(Yk)1?k?mis given by

and satisfies the QSDE

In view of(11),(12)and(14),the processesbandb#satisfy the following QSDEs(which are related to each other by conjugation):

The unitary evolution in(9)and(13)preserves the commutativity between the system and outputfield variables in the sense that

(that is,future system variables commute with the past output variables).However,the output fieldsY1,...,Ymdo not commute with each other since

and[Y(s),Y(t)T]=2imin(s,t)Jfor alls,t?0 in view of(6)and(14).The noncommutativity of the output fields makesthem inaccessible to simultaneousmeasurement.Therefore,following[38],we will consider anr-channel fieldZwhich is related tobandb#from(15)andYfrom(13)by

the first of which is equivalent toFbeing of full row rank.In view of(17),the second condition in(19)implies that[dZ,dZT]=2iF JFTdt=0,which makes the quantum processZin(18)self-commuting and allows for simultaneous continuous measurements of its entriesZ1,...,Zr.Furthermore,Zcan be regarded(up to an isomorphism)as a classical diffusion process[25]with values in Rrand a real positive definite symmetric diffusion matrixFΩFT=FFT+iF JFT=FFTin view of(5).Also,Zinherits fromYthe property(16)since

Hence,for any timet?0,any given system operator ξ(t):=f(X(t))(that is,an appropriate operator-valued extension of a complex-valued functionfto the system variables)and the past measurement history

form a set of pairwise commuting(and hence,compatible)quantum variables.This makes the processZin(18)(under the constraints(19))a legitimate model of nondemolition measurements.In what follows,we will use the conditional quantum expectation

of a system operator ξ at timet?0 with respect to the commutative von Neumann algebra Ztgenerated by the past measurement history ?tfrom(20).This is a mean square optimal estimator of ξ(t)in the sense that πt(ξ)is an element of the measurement algebra Ztwhich delivers the minimum

This characterization is similar to the variational property of classical conditional expectations(of square integrable random variables)with respect to σ-subalgebras[3].The quantum expectation Eζ =Tr(ρζ)in(22)is over the system-field density operator ρ := ω ? υ,where ω denotes the initial quantum state of the system,and υ is the vacuum state[40]of the input fields.

According to[38,Theorem 9],the conditional expectation of a given system operator ξ in(21)with respect to the nondemolition measurements(18)satisfies the Belavkin-Kushner-Stratonovich equation(BKSE)

which is driven by an innovation process χ(a martingale with respect to the measurement filtration)with the It?o differential

and diffusion matrixFFT.Here,the conditional expectation πtis evaluated entrywise at vectors of system operators,and

The drift term πt(L(ξ))dtof the SDE(23)comes from ξ having dynamics of its own in(10).The diffusion term βTKdχ represents the measurement-driven corrections of the prior estimate and,together with(24)and(25),involves additional quantities[38]which are described below for completeness.More precisely,(

Whereas the SDE(23)follows the Heisenberg picture of quantum dynamics,its dualSchr¨odingerpicture version,known as the stochastic master equation(SME)[28],describes the evolution of the posterior density operator.The latter is a quantum counterpart of the classical conditional probability distribution which is continuously updated over the course of measurements according to the Bayes rule.However,in contrast to the classical case,the SME approach is concerned with updating an operator-valued quantity rather than a usual function of several variables(such as the posterior PDF in the classical nonlinear filtering problems).

At the same time,the SDE(23)is not algebraically closed,in general,since its right-hand side involves other conditional moments which are not necessarily reducible to πt(ξ).The desired closure can be achieved within an appropriate parametric family of system operators ξ.Such family is provided,for example,by the Weyl operators(2)in the Weyl quantization framework,which is considered in the next section.

4 Evolution of the posterior quasi-characteristic function

Application of the conditional expectation(21)to the Weyloperator Wu,associated with the system variables by(2),leads to the posterior QCF

The spatial Fourier transform of(30)yields a realvalued posterior QPDF

Since the posterior QCF(30)is the conditional expectation of the Weyl operators,the BKSE(23)applies to this case too.Moreover,this leads to an algebraically closed equation for the time evolution of the posterior QCF in the framework of the Weyl quantization model for the energy operators of the system.To this end,following[50],we assume that the system Hamiltonianh0and the system-field coupling operatorsh1,...,hmin(7)are obtained by the Weyl quantization[42]of real-valued functions on Rnwith the Fourier transformsHk:Rn→C as

where Wuis the Weyl operator(2).The vectorhof the coupling operators is related to the vector-valued map

The following theorem describes the posterior QCF dynamics and employs three integral operators A,B,C which map a function φ :Rn→ C to the functions A(φ):Rn→ C and B(φ),C(φ):Rn→ Cm2as

(with the operators A and C being linear).The kernel functionV:Rn×Rn→C in(34)is computed as

whereH0andHare the Fourier transforms from(32)and(33),and Υ :Rn× Rn→ Rm×mis an auxiliary function which is expressed as

in terms of the CCR matrix Θ in(1)and the matrixJin(5).Also,the function Γ:Rn×Rn→Rm2 in(36)is related by

to the matrixEfrom(27)and(28)through the matricesE1,E2∈Rm2×m2 given by

Theorem 1Suppose the Hamiltonianh0and the coupling operatorsh1,...,hmof the quantum stochastic system(4)have the Weyl quantization form(32).Then the posterior QCF(30)with respect to the nondemolition measurements in(18),(19)satisfies the stochastic integro-differential equation(SIDE)

Here,the innovation process χ does not depend onu∈Rnand its It?o differential is given by

ProofWe will evaluate the terms of the BKSE(23)at the Weyl operator ξ:=Wuin(2)using the Weyl quantization(32)and(33).From the proof of[50,Theorem 1],it follows that the GKSL generator(7)acts on Wuas

where the functionVis computed according to(37)and(38).In view of(30),the conditional expectation of(43)takes the form

with A given by(34).The modified vectorMofcoupling operators in(26)can be represented as

where use is made of(12)and the Weyl quantization(33)of the coupling operators.In view of the WeylCCRs(1),it follows from(45)that

A similar reasoning leads to

The sum of the left-hand sides of(46)and(47)takes the form

where use is made of the matricesJ,E1,E2from(5)and(40)leading to the function Γ in(39).The conditional expectation of(48)is

with C given by(36).In particular,by lettingu=0 in(49)and recalling the property W0=IH,it follows that

Alternatively,these relations can also be obtained by applying the conditional expectation πtto the vector

whose representation employs(40)and the fact that the vectorJhin(45)consists of self-adjoint operators.Substitution of(30),(44),(49)and(50)into(23)-(25)establishes(41)and(42). ?

In the absence of measurements,the QCF Φ is no longer random.In this case,the SIDE(41)loses its diffusion term and reduces to the IDE

for the unconditional QCF obtained in[50,Theorem 1].In turn,if the system and fields are uncoupled,(51)becomes the Moyal equation[44]for the isolated system

which follows from(34)by lettingH=0 in(37).Note that both the SIDE(41)and its special cases(51)and(52)preserve the weighted positiveness of the QCF Φ mentioned at the beginning of this section.

5 Posterior QCF dynamics in the case of linear system- field coupling

We will now consider a class[50,51]of open quantum systems whose coupling operatorsh1,...,hmare linear functions of the system variables,so that

whereN∈ Rm×nis a coupling matrix,while the Hamiltonianh0consists of a quadratic part,specified by a real symmetric energy matrixRof ordern,and a nonquadratic part represented in the Weyl quantization form:

in(54)depends ond?nsystem variables comprising the vectorSX.For such a system,the representations(32)and(33)hold with

where δ′and δ′′are the distributional gradient vector and Hessian matrix of then-dimensional Dirac delta function δ.SinceSST=Id,the matrixSdescribes an isometry between Rdand the subspaceSTRd?Rn.The integral in(56),as a generalized function[69],is a complex measure on this subspace with density Ψ(with respect to thed-dimensional Lebesgue measure onSTRd).The corresponding QSDE(4)takes the form[50]

where the matricesA∈ Rn×nandB∈ Rn×mare related to the coupling and energy matricesNandRin(53)and(54)by

The nonlinear dependence on the system variables in the QSDE(58)comes from the nonquadratic part(55)of the Hamiltonian.

For example,suppose the system variables consist of the positions and momenta according to(3),and the system Hamiltonian is given by

Theorem 2Suppose the vectorhof system-field coupling operators and the system Hamiltonianh0are given by(53)and(54).Then the SIDE(41)for the posterior QCF Φ in(30)takes the form

where the matricesAandBare given by(59).The corresponding posterior QPDF ? in(31)satisfies the SIDE

where div(·)is the divergence operator with respect tox∈Rn,and the kernel function Ξ :Rn×Rd→ R is expressed as

in terms of the function Ψ and the matrixSfrom(55).Also,the It?o differential of the innovation process χ in(42)can be represented as

ProofThe drift term in(61)(and its spatial Fourier transform which is the driftterm of(62))was obtained in[50,Theorem 2]and can be established directly by substituting(56)and(57)into(37)and(34).We now turn to the diffusion terms of these SIDEs.By substituting(57)into(39)and using the relationfδ′=f(0)δ′-f′(0)δ for infinitely differentiable functionsf(see,for example,[69]),it follows that(36)takes the form

for any bounded smooth function φ :Rn→ C.In particular,atu=0,

Substitution of(65)and(66)into(35)leads to

In view of(41),application of(67)to the posterior QCF yields the diffusion term in(61)whose spatial Fourier transform leads to the diffusion term in(62).The representation(64)now follows from(42),(66)and the relation

thus completing the proof. ?

The upper line of(62)is recognizable as the Fokker-Planck-Kolmogorov equation

for the unconditional PDF of a classical Markov diffusion process with the linear driftAxand diffusion matrixBBT(see,for example,[25]).In the quantum case,the representation(68)for the posterior mean vector of the system variables,similar to the corresponding classical relations,remains valid even if the QPDF ? is not nonnegative everywhere.

Since,as mentioned in Theorem 1,the innovation process χ does not involve spatial parameters in addition to time,equations(41),(61)and(62)are simpler than stochastic partial differential equations driven by space-time white noises or more complicated random fields[70,71].Therefore,their numericalintegration can employ appropriate methods for SDEs[72].

While the above discussion was concerned with a nonlinear setting(when the function Ψ in(54)is essentially arbitrary),we will now assume that Ψ=0,so that the function Ξ in(63)vanishes,and the system is an open quantum harmonic oscillator[31].In this case,if the initial system state is Gaussian[73],the conditional quantum state remains Gaussian with the time-varying mean vector μ and the real part Σ of the quantum covariance matrix of the system variables given by

where Σ +iΘ ? 0 in view of the Heisenberg uncertainty principle[10].The corresponding Gaussian QCF is given by

Theorem 3Suppose the system dynamics are linear and specified by(53)and(54)with Ψ=0,and the initial system state is Gaussian.Then the parameters μ and Σ of the posterior Gaussian state in(69)satisfy

to the matrices(40),and the innovation process χ is driven by the measurements as

ProofBy substituting the Gaussian QCF from(70)into(61)with Ψ =0,and using the identity ?uΦμ,Σ(u)=Φμ,Σ(u)(iμ -Σu),it follows that

whereP,Qare the matrices from(73).On the other hand,application of the classical It?o lemma[25]to ln Φμ,Σ(which depends on time only through μ and Σ)yields

Here,use is made of the relation

together with the quadratic variation

of the complex-valued diffusion process in(75)and the diffusion matrixFFTof the innovation process χ.The right-hand sides of(75)and(76)are quadratic functions ofu∈Rn.By matching the corresponding coefficients,it follows that μ satisfies the SDE(71)while Σ satisfies the ODE(72).Also,(74)follows from(64)in view of(68),(69)and(73). ?

The SDE(71)and the ODE(72)are the quantum Kalman filter equations for the case of linear-Gaussian system dynamics.Similarly to the covariance dynamics ofthe usual Kalman filter[5]forclassicalsystems,(72)is organized as a differential Riccati equation(though with different matrices)which reduces to the Lyapunov ODE ˙Σ=AΣ+ΣAT+BBTin the absence of measurements.If Ψ?0 in(54),then the QSDE(58)is no longer linear,the integral operator term of the SIDE(61)comes into effect,and the Gaussian QCFs(70)can be used only as approximate solutions,which is discussed in the next section.

6 A Gaussian approximation of the posterior state

For the class ofnonlinear quantum stochastic systems with the linear coupling(53)and nonquadratic Hamiltonian(54),we will now consider a Gaussian approximation of the actual posterior quantum state of the system using the criterion

Here,?φ?is the norm in the Hilbert spaceL2(Rn)of square integrable complex-valued functions on Rn,and the minimization is over the following set of admissible parameters of the Gaussian QCF in(70):

where Sndenotes the subspace of real symmetric matrices of ordern.The squaredL2-distance in(77)is not the only proximity criterion for the actual and Gaussian quantum states.For example,[50]employs the secondorder relative Renyi entropy[74]in order to quantify the deviation of the actual QPDF ? from the Gaussian QPDFs

provided Σ ? 0(in which case,Φθin(70)is square integrable).In contrast to the relative entropy,(77)treats the actual and approximating QCFs equally and has a similar form in terms of the QPDFs(31)and(79)due to the Plancherel identity:

Furthermore,a different approximation is provided by the quantum Gaussian stochastic linearization of the governing QSDE through the quadratic approximation of the system Hamiltonian[75].

If the actual posterior QCF Φ(or the corresponding posterior QPDF ?)were known,then the parameter θ could be chosen,at every moment of time,so as to minimize the cost in(77)(or equivalently,(80)).However,in the nonlinear case Ψ≠0,when the actual posterior QCF and QPDF are difficult to find,θ can be evolved“along”the orthogonal projection(in theL2sense)of the Ito differential dΦ of the random field Φ from(61)(whose right-hand side is computed atΦ = Φθ)onto the tangent space of differentials dΦθof the Gaussian QCFs in(76),provided θ belongs to the interior of the set?in(78).This approach(whose general idea is similar to the projective filtering in[76]and references therein)leads to a modified version of the quantum Kalman filter equations(71)and(72):

These equations involve additional terms λ∈Rnand σ∈Snwhich are found as a solution to the minimization problem

denotes the negative of the integral operator term in(61)which is contributed by the nonquadratic part of the system Hamiltonian.Both functions?μ,Σ(λ,σ,u)and?μ,Σ(u)are Hermitian with respect tou∈ Rn.The following theorem computes the correction terms λ and σ.

Theorem 4Suppose θ is an interior point of the set?in(78),and Σ?0.Then the minimum in(83)is achieved at a unique point

where SΣis a positive definite self-adjoint operator on Sngiven by

where 〈·,·〉denotes the Frobenius inner product of matrices.

ProofFor what follows,the set Rn×Snis regarded as a Hilbert space with the direct-sum inner product〈(μ1,Σ1),(μ2,Σ2)〉Rn×Sn:= μT1μ2+Tr(Σ1Σ2)generated from the Euclidean inner product in Rnand the Frobenius inner product of matrices in Sn.The function being minimized in(83)is a convex quadratic function of(λ,σ)∈ Rn× Sn,whose Frechet differentiation leads to the necessary conditions of optimality:

where use is made of the relations

which follow from(84).Note that

Here,the operator SΣis given by(88)and originates from the relation

foranyσ∈Snand an Rn-valued Gaussian random vector ξ with zero mean and covariance matrixC.This follows from the Isserlis theorem[77,78]applied to the fourthorder mixed moments of the entries of ξ.The moments of arbitrary odd order for such a vector vanish.Positive definiteness(and hence,invertibility)of the operator SΣfollows from the inequalities

the second of which holds for any σ∈Sn{0}.Now,a combination of(89)and(92)with(85)allows λ to be uniquely found as

where use is made of the identity

for the Gaussian QCFs(70)together with the momentgenerating function

Substitution of the Hessian matrix

of the function(95)into(96)leads to(87). ?

The following theorem describes the computation of the inverse operator S-1Σwhich is required for the evaluation of(87).

Theorem 5For any positive definite matrix Σ∈Sn,the inverse of the operator SΣin(88)can be computed as

ProofSince Σ ? 0,the operator SΣcan be represented as the composition

where I is the identity operator on Sn.Here,TΣis a positive definite self-adjoint operator acting on a matrix σ∈Snas

The idempotence Π2= Π of the projection operator implies that

A combination of(98)with(101)leads to the representation

In combination with(97),equations(86)and(87)provide integral representations of the correction terms λ and σ in the modified quantum Kalman filter(81),(82)as nonlinear functions of μ and Σ.These integrals involve the spatialFouriertransformΨ ofthe nonquadratic part of the Hamiltonian.Their closed-form evaluation is possible,for example,if Ψ is a linear combination of quadratic-exponential functions(see[50,Section 9]),which corresponds to the presence of Gaussian-shaped“bumps”in the potential energy of the system[79];cf.(60).This consideration can be used in order to apply the above results to open quantum systems with multiextremum energy landscapes.However,the circle of questions in regard to the above described Gaussian approximation includes its error analysis and the study of conditions when(82)produces a physically meaningful matrix Σ satisfying the Heisenberg uncertainty principle Σ+iΘ ?0.

7 Conclusions

For a class of quantum stochastic systems,whose Hamiltonian and coupling operators are represented in the Weyl quantization form,we have obtained a nonlinear SIDE for the evolution of the posterior QCF conditioned on multichannel nondemolition measurements.This equation is driven by a classical diffusion process of innovations associated with the measurements.We have also considered a more specific form of the SIDE for the case of linear system-field coupling and outlined a Gaussian approximation of the posterior state governed by modified quantum Kalman filter equations.These ideas are applicable to the development of suboptimal quantum filtering algorithms which employ more complicated(for example,multi-Gaussian)approximations of the posterior QCF and QPDF.Furthermore,the results of this paper can be extended to more general system dynamics,field states and measurementsettings(such as nonlinear coupling,coherent and Gaussian states and photon counting measurements),for some of which the BKSE was considered in[36-38]without using the Weylquantization ofthe Hamiltonian and coupling operators.

Acknowledgements

The author thanks the anonymous reviewers for useful comments.

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31 January 2017;revised 5 May 2017;accepted 5 May 2017

DOI 10.1007/s11768-017-7012-2

E-mail:igor.g.vladimirov@gmail.com.

This paper is dedicated to Professor Ian R.Petersen on the occasion of his 60th birthday.This work was initiated while the author was with the UNSW Canberra,Australia,where it was supported by the Australian Research Council,and was completed at the Australian National University under support of the Air Force Office of Scientific Research(AFOSR)under agreement number FA2386-16-1-4065.A brief version[80]of this paper was presented at the IEEE 2016 Conference on Norbert Wiener in the 21st Century,13-15 July 2016,Melbourne,Australia.

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Igor G.VLADIMIROVreceived M.Sc.degree in Control Systems in 1989 and Ph.D.degree in Physics and Mathematics(with specialization in Mathematical Cybernetics)in 1992 from the DepartmentofControland Applied Mathematics of the Moscow Institute(State University)of Physics and Technology,Russia.He worked as a Senior Research Associate at the State Research Institute ofAviation Systems in 1993-1997 and the Institute for Information Transmission Problems,the Russian Academy of Sciences,Moscow,in 1994-2007,in the areas of applied Kalman filtering,stochastic robust control,hysteresis systems and spatially discretized dynamical systems.From 1997 to 2008,Dr.Vladimirov held research academic positions at the Mathematics Department and School of Engineering at the University of Queensland,Brisbane,Australia,working in the above areas and in stochastic modelling of econometric time series,lattice models of statistical mechanics and transport phenomena in random media.In 2000,he also had a visiting research fellowship at the School of Mathematical Sciences,Queen Mary and Westfield College,University of London,working onp-adic analysis of Hamiltonian roundoff.From 2009 to 2016,Dr.Vladimirov was a Senior Research Fellow at the University of New South Wales Canberra,doing research on quantum stochastic filtering and control,which he continues after moving to the Australian National University in 2017.In 2013,Dr.Vladimirov was awarded B.N.Petrov prize of the Russian Academy of Sciences for his works on the anisotropy-based theory of stochastic robust filtering and control.E-mail:igor.g.vladimirov@gmail.com.