Direct coupling coherent quantum observers with discounted mean square performance criteria and penalized back-action
DDirect Coupling Coherent Quantum Observers with DiscountedMean Square Performance Criteria and Penalized Back-action (cid:63)
Igor G. Vladimirov a , Ian R. Petersen a a Australian National University, Canberra, Australia
Abstract
This paper is concerned with quantum harmonic oscillators consisting of a quantum plant and a directly coupled coherent quantumobserver. We employ discounted quadratic performance criteria in the form of exponentially weighted time averages of second-ordermoments of the system variables. Small-gain-theorem bounds are obtained for the back-action of the observer on the covariance dynamicsof the plant in terms of the plant-observer coupling. A coherent quantum filtering (CQF) problem is formulated as the minimization ofthe discounted mean square of an estimation error, with which the dynamic variables of the observer approximate those of the plant. Thecost functional also involves a quadratic penalty on the plant-observer coupling matrix in order to mitigate the back-action effect. For thediscounted mean square optimal CQF problem with penalized back-action, we establish first-order necessary conditions of optimality inthe form of algebraic matrix equations. By using the Hamiltonian structure of the Heisenberg dynamics and Lie-algebraic techniques, thisset of equations is represented in a more explicit form for equally dimensioned plant and observer. For a class of such observers withautonomous estimation error dynamics, we obtain a solution of the CQF problem and outline a homotopy method. The computation ofthe performance criteria and the observer synthesis are illustrated by numerical examples.
Key words:
Quantum harmonic oscillator; direct coupling; coherent quantum filtering; observer back-action; discounted mean squareoptimality; Hamiltonian matrices; Lie algebra.
Noncommutative counterparts of classical control and fil-tering problems [1,2,25] are a subject of active research inquantum control which is concerned with dynamical andstochastic systems governed by the laws of quantum me-chanics and quantum probability [19,30]. These develop-ments (see, for example, [23,31,33,52,53]) are particularlyfocused on open quantum systems whose internal dynam-ics are affected by interaction with the environment [8]. Insuch systems, the evolution of dynamic variables (as non-commutative operators on a Hilbert space) is often modelledusing the Hudson-Parthasarathy calculus [18,21,35] whichprovides a rigorous framework of quantum stochastic differ-ential equations (QSDEs) driven by quantum Wiener pro-cesses on symmetric Fock spaces. In particular, linear QS-DEs model open quantum harmonic oscillators (OQHOs)[13] whose dynamic variables (such as the position and mo-mentum or annihilation and creation operators [29,42]) sat-isfy canonical commutation relations (CCRs). This class of (cid:63)
This work is supported by the Air Force Office of ScientificResearch (AFOSR) under agreement number FA2386-16-1-4065and the Australian Research Council under grant DP180101805.A brief version [56] of this paper was presented at the IEEE 2016Conference on Norbert Wiener in the 21st Century.
Email addresses: [email protected] (Igor G. Vladimirov), [email protected] (Ian R.Petersen).
QSDEs is important for linear quantum control theory [37]and applications to quantum optics [15,58] which providesone of platforms for quantum information technologies [32].One of the fundamental problems for quantum stochasticsystems is the coherent quantum linear quadratic Gaussian(CQLQG) control problem [33] which is a quantum me-chanical counterpart of the classical LQG control problem.The latter is well-known in linear stochastic control theorydue to the separation principle and its links with Kalmanfiltering and deterministic optimal control settings such asthe linear quadratic regulator (LQR) problem [2,25]. Co-herent quantum feedback control [26,61] employs the ideaof control by interconnection, whereby quantum systemsinteract with each other directly or through optical fieldsin a measurement-free fashion, which can be described us-ing the quantum feedback network formalism [16]. In com-parison with the traditional observation-actuation controlparadigm, coherent quantum control avoids the “lossy” con-version of operator-valued quantum variables into classicalsignals (which underlies the quantum measurement process),is potentially faster and can be implemented on micro andnano-scales using natural quantum mechanical effects.In coherent quantum filtering (CQF) problems [31,53],which are “feedback-free” versions of the CQLQG controlproblem, an observer is cascaded in a measurement-freefashion with a quantum plant so as to develop quantumcorrelations with the latter over the course of time. Both
Preprint submitted to Automatica Wednesday 5 th September, 2018 a r X i v : . [ c s . S Y ] S e p roblems employ mean square performance criteria andinvolve physical realizability (PR) constraints [23,44] onthe state-space matrices of the quantum controllers and fil-ters. The PR constraints are a consequence of the specificHamiltonian structure of quantum dynamics and compli-cate the design of optimal coherent quantum controllersand filters. Variational approaches of [51]–[53] reformulatethe underlying problem as a constrained covariance controlproblem and employ an adaptation of ideas from dynamicprogramming, the Pontryagin minimum principle [41,48]and nonlinear functional analysis. In particular, the Frechetdifferentiation of the LQG cost with respect to the state-space matrices of the controller or filter subject to the PRconstraints leads to necessary conditions of optimality inthe form of nonlinear algebraic matrix equations. Althoughthis approach is quite similar to [4,47] (with the quantumnature of the problem manifesting itself only through thePR constraints), the resulting equations appear to be muchharder to solve than their classical predecessors.Fully quantum variational techniques, using perturbationanalysis [45,54,55,57] beyond the class of OQHOs and sym-plectic geometric tools [46], suggest that the complicatedsets of nonlinear equations for optimal quantum controllersand filters may appear to be more amenable to solution ifthey are approached using Hamiltonian structures similar tothose in the underlying quantum dynamics. Such structuresare particularly transparent in closed QHOs. Indeed, thesemodels of linear quantum systems do not involve externalbosonic fields and are technically simpler than the abovementioned OQHOs (although leave room for modelling thelatter, for example, through the Caldeira-Leggett infinite sys-tem limit using a bath of harmonic oscillators [9]).We employ this class of models in the present paper andconsider a mean square optimal CQF problem for a plantand a directly coupled observer which form a closed QHO.Since this setting does not use quantum Wiener processes,it simplifies the technical side of the treatment in compar-ison with [31,53]. The Hamiltonian of the plant-observerQHO is a quadratic function of the dynamic variables satis-fying the CCRs. When the energy matrix, which specifies thequadratic form of the Hamiltonian, is positive semi-definite,the system variables of the QHO are either constant or ex-hibit oscillatory behaviour. This motivates the use of a costfunctional (being minimized) in the form of a discountedmean square of an estimation error (with an exponentiallydecaying weight [7]) with which the observer variables ap-proximate given linear combinations of the plant variables ofinterest. The performance criterion also involves a quadraticpenalty on the plant-observer coupling in order to achieve acompromise between the conflicting requirements of mini-mizing the estimation error and reducing the back-action ofthe observer on the plant. The CQF problem with penalizedback-action can also be regarded as a quantum-mechanicalcounterpart to the classical LQR problem. The use of dis-counted averages of nonlinear moments of system variablesand the presence of optimization makes this setting differ-ent from the time-averaged approach of [38,39] to CQF indirectly coupled QHOs (see [40] for a quantum-optical im-plementation of that approach). Since discounted moments of system variables for QHOsplay an important role throughout the paper, we discuss thecomputation of such moments in the state-space and fre-quency domains for completeness. Using the ideas of thesmall-gain theorem (see, for example, [11] and referencestherein) and linear matrix inequalities, we establish upperbounds for the back-action of the observer on the covari-ance dynamics of the plant in terms of the plant-observercoupling. This leads to a lower bound for the mean squareof the estimation error in terms of its value for uncoupledplant and observer. Similarly to the variational approach of[52,53], we develop first-order necessary conditions of op-timality for the CQF problem being considered. These con-ditions are organized as a set of two algebraic Lyapunovequations (ALEs) for the controllability and observabilityGramians which are coupled through another equation forthe Hankelian (the product of the Gramians) of the plant-observer composite system. The Hamiltonian structure ofthe underlying Heisenberg dynamics allows Lie-algebraictechniques (in particular, the Jacobi identity [12]) to be em-ployed in order to represent this set of equations in termsof the commutators of appropriately transformed Gramians.This leads to a more tractable form of the optimality condi-tions for equally dimensioned plant and observer. We singleout a class of such observers with autonomous estimationerror dynamics, for which the CQF problem is amenableto numerical solution through a homotopy method (similarto [28]) over the penalty parameter. We also investigate theasymptotic behaviour of the resulting optimal observers inthe weak-coupling limit, and illustrate the performance cri-teria computation and observer synthesis by numerical ex-amples.The paper is organised as follows. Section 2 specifies theclosed QHOs including its subclass with positive semi-definite energy matrices. Section 3 describes the discountedaveraging of moments for system operators in such QHOsin the time and frequency domains and illustrates theircomputation by a numerical example. Section 4 speci-fies the direct coupling of quantum plants and coherentquantum observers. Section 5 discusses bounds for the ob-server back-action on the covariance dynamics of the plant.Section 6 formulates the discounted mean square optimalCQF problem with penalized back-action and discussescoupling-estimation inequalities. Section 7 establishes first-order necessary conditions of optimality for this problem.Section 8 represents the optimality conditions in a Lie-algebraic form. Section 9 provides a suboptimal solution ofthe CQF problem for a class of observers with autonomousestimation error dynamics and gives a numerical example ofobserver synthesis. Section 10 makes concluding remarks. Consider a QHO [29] with an even number n of dynamicvariables X , . . . , X n which are time-varying self-adjoint op-erators on a complex separable Hilbert space H satisfying2he CCRs [ X ( t ) , X ( t ) T ] : = ([ X j ( t ) , X k ( t )]) (cid:54) j , k (cid:54) n = i Θ , X : = X ( t ) ... X n ( t ) (1)at any instant t (cid:62) CCR matrix Θ ∈ A n is nonsingular. Here, A n denotes the subspace of real anti-symmetric matrices of order n . The entries θ jk of Θ in (1)represent the scaling operators θ jk I , with I the identityoperator on H . The transpose ( · ) T acts on matrices of op-erators as if the latter were scalars, vectors are organized ascolumns unless indicated otherwise, [ φ , ψ ] : = ϕψ − ψϕ isthe commutator of operators, and i : = √− H : = X T RX , (2)specified by an energy matrix R ∈ S n , with S n the subspaceof real symmetric matrices of order n . Due to (1) and (2), theHeisenberg dynamics of the QHO are governed by a linearODE ˙ X = i [ H , X ] = AX , (3)where A ∈ R n × n is a matrix of constant coefficients given by A : = Θ R . (4)The solution of the ODE (3) is expressed using the standardmatrix exponential as X ( t ) = j t ( X ) : = U ( t ) † X U ( t ) = e it ad H ( X ) = e tA X , (5)where ad α : = [ α , · ] , and the subscript ( · ) indicates the ini-tial values at time t =
0. The first three equalities in (5)apply to a general Hamiltonian H (that is, not necessarilya quadratic function of X ), and U ( t ) : = e − itH is a time-varying unitary operator on H (with the adjoint U ( t ) † = e itH ), which specifies the flow j t in (5) acting as a uni-tary similarity transformation on the system variables. Theflow j t preserves the CCRs (1) which, in view of the rela-tion [ X ( t ) , X ( t ) T ] = e tA [ X , X T0 ] e tA T = i e tA Θ e tA T = i Θ , areequivalent to the symplectic property e tA Θ e tA T = Θ of thematrix e tA for any time t (cid:62)
0. The infinitesimal form of thisproperty is A Θ + Θ A T =
0. This equality corresponds to thePR conditions for OQHOs [23,43] and its fulfillment is en-sured by the Hamiltonian structure A ∈ Θ S n of the matrix A in (4). Similarly to classical linear systems, if the initialquantum state of the QHO is Gaussian [10,36], it remainsso over the course of time due to the deterministic lineardependence of X ( t ) on X in (5). If the energy matrix in(2) is positive semi-definite, R (cid:60) √ R (cid:60) A = Θ √ R √ R is isospectral to the ma-trix 2 √ R Θ √ R ∈ A n whose eigenvalues are purely imagi-nary [20]. In the case R (cid:31)
0, this follows directly from thesimilarity transformation A = R − / ( √ R Θ √ R ) √ R (6) (see, for example, [38]), whereby A is diagonalized as A = iV Ω W , W : = V − , Ω : = diag (cid:54) k (cid:54) n ( ω k ) . (7)Here, W : = ( w jk ) (cid:54) j , k (cid:54) n ∈ C n × n is the inverse of anonsingular matrix V : = ( v jk ) (cid:54) j , k (cid:54) n ∈ C n × n whosecolumns V , . . . , V n ∈ C n are the eigenvectors of A , and Ω : = diag (cid:54) k (cid:54) n ( ω k ) ∈ R n × n is a diagonal matrix of fre-quencies of the QHO. These frequencies (which should notbe confused with the eigenvalues of the Hamiltonian H asan operator on H describing the energy levels of the QHO[42]) are nonzero and symmetric about the origin, and,without loss of generality, are assumed to be arranged so that ω k = − ω k + n > , k = , . . . , n . (8)Note that √ RV is a unitary matrix whose columns are theeigenvectors of the matrix i √ R Θ √ R ∈ H n in view of (6); seealso the proof of Williamson’s symplectic diagonalizationtheorem [59,60] in [12, pp. 244–245]. Here, H n denotes thesubspace of complex Hermitian matrices of order n . Substi-tution of (7) into (5) leads to X ( t ) = V e it Ω W X . (9)Due to the presence of the matrix e it Ω = diag (cid:54) k (cid:54) n ( e i ω k t ) in(9), the dynamic variables of the QHO are linear combina-tions of their initial values whose coefficients are trigono-metric polynomials of time: X j ( t ) = n ∑ k ,(cid:96) = c jk (cid:96) e i ω k t X (cid:96) ( ) , j = , . . . , n , (10)where c jk (cid:96) are complex parameters which are assembled intorank-one matrices C k : = ( c jk (cid:96) ) (cid:54) j ,(cid:96) (cid:54) n = V k W k , c jk (cid:96) : = v jk w k (cid:96) , (11)with W k denoting the k th row of W . The matrices C , . . . , C n form a resolution of the identity: ∑ nk = C k = VW = I n . Also, C k = C k + n , k = , . . . , n , (12)in accordance with (8), whereby (10) can be represented invector-matrix form as X ( t ) = n / ∑ k = (cid:0) e i ω k t C k + e − i ω k t C k (cid:1) X = n / ∑ k = Re ( e i ω k t C k ) X , (13)where ( · ) is the complex conjugate. Therefore, for any posi-tive integer d and any d -index j : = ( j , . . . , j d ) ∈ { , . . . , n } d ,the following degree d monomial of the system variables isalso a trigonometric polynomial of time t : Ξ j ( t ) : = d −→ ∏ s = X j s ( t ) = ∑ k ,(cid:96) ∈{ ,..., n } d d ∏ s = c j s k s (cid:96) s e i ω ks t Ξ (cid:96) ( ) . (14)3ere, −→ ∏ denotes the “rightwards” ordered product ofoperators (the order of multiplication is essential for non-commutative quantum variables), and the sum is taken over d -indices k : = ( k , . . . , k d ) , (cid:96) : = ( (cid:96) , . . . , (cid:96) d ) ∈ { , . . . , n } d .Note that (10) is a particular case of (14) with d =
1. Therelations (9)–(14) remain valid in the case of R (cid:60)
0, exceptthat (8) is relaxed to the frequencies ω , . . . , ω n / beingnonnegative. For any τ >
0, we define a linear functional E τ which maps asystem operator σ of the QHO to the weighted time average E τ σ : = τ (cid:90) + ∞ e − t / τ E σ ( t ) d t . (15)Here, E σ : = Tr ( ρσ ) denotes the quantum expectation overthe underlying quantum state ρ (which is a positive semi-definite self-adjoint operator on H with unit trace). Theweighting function τ e − t / τ in (15) is the density of an ex-ponential probability distribution with mean value τ . There-fore, τ plays the role of an effective horizon for averaging E σ over time. This time average (where the relative im-portance of the quantity of interest decays exponentially)has the structure of a discounted cost functional in dy-namic programming problems [7]. In particular, if E σ ( t ) ,as a function of time t (cid:62)
0, is right-continuous at t = τ → + E τ σ = E σ . At the other extreme, the infinite-horizon average of σ is defined by E ∞ σ : = lim τ → + ∞ E τ σ = lim τ → + ∞ (cid:16) τ (cid:90) τ E σ ( t ) d t (cid:17) , (16)provided these limits exist. The second of these equalities,whose right-hand side is the Cesaro mean of E σ , followsfrom the integral version of the Hardy-Littlewood Taube-rian theorem [14]. In particular, (16) implies that | E ∞ σ | (cid:54) lim sup t → + ∞ | E σ ( t ) | .In the case when the QHO has a positive semi-definite energymatrix, the coefficients in (13) and (14) are either constantor oscillatory, which makes the time averages (15) and (16)well-defined for nonlinear functions of the system variablesand their moments for any τ >
0. A similar property under-lies applications of harmonic analysis to the heterodyne de-tection of signals. To this end, we will use the characteristicfunction χ τ : R → C of the exponential distribution and itspointwise convergence: χ τ ( u ) : = τ (cid:90) + ∞ e − t / τ e iut d t = − iu τ → δ u = (cid:40) u =
00 if u (cid:54) = , as τ → + ∞ , (17)where δ pq is the Kronecker delta. A combination of (14)with (17) implies that if the initial system variables of theQHO have finite mixed moments E Ξ (cid:96) ( ) of order d for all (cid:96) ∈ { , . . . , n } d , then such moments have the following time-averaged values (15): E τ Ξ j : = τ (cid:90) + ∞ e − t / τ E Ξ j ( t ) d t = ∑ k ∈{ ,..., n } d χ τ (cid:16) d ∑ s = ω k s (cid:17) ∑ (cid:96) ∈{ ,..., n } d d ∏ s = c j s k s (cid:96) s E Ξ (cid:96) ( ) (18)for any j ∈ { , . . . , n } d . Hence, the corresponding infinite-horizon average (16) takes the form E ∞ Ξ j = ∑ k ∈ K d ∑ (cid:96) ∈{ ,..., n } d d ∏ s = c j s k s (cid:96) s E Ξ (cid:96) ( ) , (19)where K d : = (cid:8) ( k , . . . , k d ) ∈ { , . . . , n } d : ∑ ds = ω k s = (cid:9) is a subset of d -indices associated with the frequencies ω , . . . , ω n of the QHO from (7). For every even d , theset K d is nonempty due to the central symmetry of thefrequencies. If the QHO is in a Gaussian quantum state,mentioned in Section 2, then the higher-order momentson the right-hand sides of (18) and (19) can be expressedin terms of the first two moments (with d (cid:54)
2) by usingthe Isserlis-Wick theorem [22,24]. However, the Gaussianassumption will not be employed in what follows.The linear functional E τ in (18) and its limit E ∞ in (19)are extendable to polynomials and more general functions σ : = f ( X ) of the system variables, provided X satisfies ap-propriate integrability conditions. Such an extension of E ∞ ,which involves the Cesaro mean, is similar to the argumentused in the context of Besicovitch spaces of almost periodicfunctions [5]. If the system is in an invariant state ρ (which,therefore, satisfies [ H , ρ ] = E σ = Tr ( ρ e it ad H ( σ )) = Tr ( e − it ad H ( ρ ) σ ) = Tr ( ρσ ) is time-independent for any system operator σ evolved bythe flow (5). In this case, the time averaging in (15) becomesredundant. However, the subsequent discussion is concernedwith general (not necessarily invariant) quantum states ρ .The following theorem is, in essence, an adaptation of clas-sical results on the averaging of quasi-periodic motions inHamiltonian systems; see, for example, [3, pp. 285–289]. Theorem 1
Suppose the energy matrix in (2) satisfies R (cid:31) . Also, let the frequencies ω , . . . , ω n / of the QHO, ar-ranged according to (8), be incommensurable in the senseof rational independence (that is, their linear combination ∑ n / k = λ k ω k with integer coefficients λ , . . . , λ n / ∈ Z vanishesif and only if λ = . . . = λ n / = ). Furthermore, suppose afunction f : R n → R is extended to quantum variables so thatg ( ϕ ) : = E f (cid:16) n / ∑ k = Re ( e i ϕ k C k ) X (cid:17) (20) depends continuously on the phases ϕ : = ( ϕ k ) (cid:54) k (cid:54) n / ∈ T n / , where T is the one-dimensional torus implemented as he interval [ , π ) , and the matrices C k are given by (11).Then the system operator f ( X ) has the following infinite-horizon average value (16): E ∞ f ( X ) = ( π ) − n / (cid:90) T n / g ( ϕ ) d ϕ . (21) Proof.
From (13) and (20), it follows that E f ( X ( t )) = g ( t (cid:102) ) for any t (cid:62)
0, where (cid:102) : = ( ω k ) (cid:54) k (cid:54) n / , and the entriesof the vector t (cid:102) ∈ R n / are considered modulo 2 π . Sincethe function g , which is 2 π -periodic in each of its vari-ables, is assumed to be continuous (and hence, g isbounded due to the compactness of the torus), then (21)is established as E ∞ f ( X ) = lim τ → + ∞ (cid:16) τ (cid:82) τ g ( t (cid:102) ) d t (cid:17) =( π ) − n / (cid:82) T n / g ( ϕ ) d ϕ . The last equality is obtained byapplying the Weyl equidistribution criterion [6] to themap t (cid:55)→ t (cid:102) considered modulo 2 π , whereby its sampledistribution D τ ( S ) : = τ µ { (cid:54) t (cid:54) τ : t (cid:102) ∈ S + Z n / } for S ⊂ T n / converges weakly to the uniform proba-bility measure ( π ) − n / µ n / ( S ) on the torus T n / , pro-vided the frequencies are incommensurable. More pre-cisely, lim τ → + ∞ D τ ( S ) = ( π ) − n / µ n / ( S ) for any Borel set S ⊂ T n / whose boundary ∂ S satisfies µ n / ( ∂ S ) =
0, where µ r denotes the r -dimensional Lebesgue measure. (cid:4) The vectors (cid:102) of commensurable frequencies are con-tained in a denumerable union (cid:83) λ ∈ Z n / \{ } λ ⊥ of the hy-perplanes λ ⊥ : = { (cid:102) ∈ R n / : λ T (cid:102) = } which has zero n / τ and does not employ theimaginarity of the spectrum of A and the frequency incom-mensurability condition of Theorem 1. To this end, we notethat E ( XX T ) ∈ H + n at every moment of time due to the gen-eralized Heisenberg uncertainty principle [19], where H + n denotes the set of complex positive semi-definite Hermitianmatrices of order n . Furthermore, Im E ( XX T ) = Θ remainsunchanged in view of the preservation of the CCRs (1) men-tioned above. Also, with any Hurwitz matrix α , we asso-ciate a linear operator L ( α , · ) which maps an appropriatelydimensioned matrix β to a unique solution γ = L ( α , β ) ofthe ALE αγ + γα T + β = L ( α , β ) : = (cid:90) + ∞ e t α β e t α T d t . (22)The monotonicity of the operator L ( α , · ) (with respect tothe partial ordering induced by positive semi-definiteness)implies that L ( α , L ( α , β )) = L ( α , (cid:112) β β − / L ( α , β ) β − / (cid:112) β ) (cid:52) r ( L ( α , β ) β − ) L ( α , β ) (23) for any β (cid:31)
0, where r ( · ) denotes the spectral radius ofa matrix, and use is made of the similarity transformation β − / N β − / (cid:55)→ N β − . Lemma 2
Let the initial dynamic variables of the QHOhave finite second moments (that is, E ( X T0 X ) < + ∞ ) whosereal parts form the matrix Σ : = Re E ( X X T0 ) . (24) Also, suppose the effective time horizon τ > is boundedabove as τ < ( , ln r ( e A )) . (25) Then the matrix of the real parts of the discounted secondmoments of the dynamic variables can be computed asP : = Re E τ ( XX T ) = τ L ( A τ , Σ ) (26) through the operator (22). That is, P is a unique solution ofthe ALE A τ P + PA T τ + τ Σ = , (27) with the Hurwitz matrixA τ : = A − τ I n . (28) Proof.
By combining (5) with (24), it follows thatRe E ( X ( t ) X ( t ) T ) = e tA Σ e tA T for any t (cid:62)
0. Hence, in ap-plication to the matrix P in (26), the time average (15)can be computed as P = τ (cid:82) + ∞ e − t / τ Re E ( X ( t ) X ( t ) T ) d t = τ (cid:82) + ∞ e − t / τ e tA Σ e tA T d t = τ (cid:82) + ∞ e tA τ Σ e tA T τ d t = τ L ( A τ , Σ ) ,thus establishing the representation (26). Here, the matrix A τ , given by (28), is Hurwitz due to the condition (25). (cid:4) In view of (27), the matrix P is the controllability Gramian[25] of the pair ( A τ , √ τ − Σ ) . In contrast to similar ALEs forsteady-state covariance matrices in dissipative OQHOs [13](where the corresponding matrix A itself is Hurwitz), theterm τ Σ in (27) comes from the initial condition (24) insteadof the Ito matrix of the quantum Wiener process [19,21,35].Since A is a Hamiltonian matrix (and hence, its spectrumis symmetric about the imaginary axis), the condition (25)is equivalent to the eigenvalues of A being contained in thestrip (cid:8) z ∈ C : | Re z | < τ (cid:9) . For any τ > P in (26) is P = πτ Re (cid:90) + ∞ − ∞ F (cid:16) τ + i ω (cid:17) Γ F (cid:16) τ + i ω (cid:17) ∗ d ω = πτ Im (cid:90) Re s = τ F ( s ) Γ F ( s ) ∗ d s , (29)with ( · ) ∗ : = (( · )) T the complex conjugate transpose. Here, Γ : = E ( X X T0 ) = Σ + i Θ (30)is the matrix of second moments of the initial system vari-ables, and F ( s ) : = ( sI n − A ) − (31)5s the transfer function (with the complex variable s satisfy-ing Re s > ln r ( e A ) ) which relates the Laplace transform (cid:101) X ( s ) : = (cid:90) + ∞ e − st X ( t ) d t (32)of the quantum process X from (5) to its initial value X as (cid:101) X ( s ) = (cid:82) + ∞ e − t ( sI n − A ) d tX = F ( s ) X . The representa-tion (29) is obtained by applying an operator version ofthe Plancherel theorem to the inverse Fourier transforme − t τ X ( t ) = π (cid:82) + ∞ − ∞ e i ω t (cid:101) X (cid:0) τ + i ω (cid:1) d ω for t (cid:62) R (cid:60) A in (4) has a purely imag-inary spectrum and (25) holds for any arbitrarily large τ ), theformal limit of the ALE (27), as τ → + ∞ , is AP + PA T = E ∞ ( XX T ) . We will therefore provide an alternativecalculation of the discounted second moments for complete-ness. Lemma 3
Suppose the energy matrix R of the QHO in (2)satisfies R (cid:31) , and the initial dynamic variables have finitesecond moments assembled into the matrix Γ in (30). Thenfor any τ > , E τ ( XX T ) = V ( Φ τ (cid:12) ( W Γ W ∗ )) V ∗ = n / ∑ j , k = (cid:104) V j V j (cid:105) (cid:32)(cid:34) χ τ ( ω j − ω k ) χ τ ( ω j + ω k ) χ τ ( − ω j − ω k ) χ τ ( ω k − ω j ) (cid:35) (cid:12) (cid:16)(cid:34) W j W j (cid:35) Γ (cid:104) W ∗ k W T k (cid:105)(cid:17)(cid:33) (cid:34) V ∗ k V T k (cid:35) . (33) Here, V is the matrix from (7), use is made of an auxiliarymatrix Φ τ : = ( χ τ ( ω j − ω k )) (cid:54) j , k (cid:54) n (34) associated with the frequencies of the QHO through the func-tion χ τ from (17), (cid:12) denotes the Hadamard product of ma-trices [20], and C k are the matrices from (11) satisfying (12)under the convention (8). Furthermore, the infinite-horizontime averages of the second moments are computed as E ∞ ( XX T ) = V ( Φ ∞ (cid:12) ( W Γ W ∗ )) V ∗ = n / ∑ j , k = δ ω j ω k (cid:0) C j Γ C ∗ k + C j Γ C T k (cid:1) = n / ∑ j , k = δ ω j ω k (cid:104) V j V j (cid:105)(cid:34) W j Γ W ∗ k W j Γ W T k (cid:35)(cid:34) V ∗ k V T k (cid:35) , (35) where use is made of a binary matrix Φ ∞ : = ( δ ω j ω k ) (cid:54) j , k (cid:54) n . (36) Proof.
Although (33) can be obtained from the relation (18)with d =
2, we will provide a direct calculation. In view of self-adjointness of the system variables, (9) and (13) implythat X ( t ) X ( t ) T = X ( t ) X ( t ) † = V e it Ω W X X T0 W ∗ e − it Ω V ∗ = V (cid:0) Ψ ( t ) (cid:12) ( W X X T0 W ∗ ) (cid:1) V ∗ = n / ∑ j , k = (cid:0) e i ω j t C j + e − i ω j t C j (cid:1) X X T0 (cid:0) e − i ω k t C ∗ k + e i ω k t C T k (cid:1) , (37)with ( · ) † : = (( · ) ) T the transpose of the entry-wise operatoradjoint ( · ) . Here, use is also made of the diagonal structureof the matrix Ω in (7) together with a complex Hermitianrank-one matrix Ψ ( t ) : = ( e i ( ω j − ω k ) t ) (cid:54) j , k (cid:54) n (38)which encodes the time dependence of XX T . The represen-tation (37) allows the time averaging to be decoupled fromthe quantum expectation as E τ ( XX T ) : = τ (cid:90) + ∞ e − t / τ E ( X ( t ) X ( t ) T ) d t = V (cid:16) τ (cid:90) + ∞ e − t / τ Ψ ( t ) d t (cid:12) ( W Γ W ∗ ) (cid:17) V ∗ = n / ∑ j , k = (cid:16) χ τ ( ω j − ω k ) C j Γ C ∗ k + χ τ ( ω j + ω k ) C j Γ C T k + χ τ ( − ω j − ω k ) C j Γ C ∗ k + χ τ ( ω k − ω j ) C j Γ C T k (cid:17) , (39)which leads to (33) and (34) in view of (11). Here, Γ is thematrix given by (30), and the relation τ (cid:82) τ e − t / τ Ψ ( t ) d t = Φ τ is obtained by applying (17) entrywise to the matrix Ψ in (38). Now, the convergence in (17) implies that thematrices (34) and (36) are related by lim τ → + ∞ Φ τ = Φ ∞ ,and lim τ → + ∞ χ τ ( ± ( ω j + ω k )) = ω j + ω k > j , k = , . . . , n in view of (8). This leads to (35) in view of(39). (cid:4) The proof of Lemma 3 shows that E τ ( XX T ) is close to E ∞ ( XX T ) if the effective time horizon τ is large in compar-ison with τ ∗ : = (cid:0)(cid:8) | ω j ± ω k | : 1 (cid:54) j , k (cid:54) n (cid:9) \{ } (cid:1) . (40)If the frequencies ω , . . . , ω n / are pairwise different (whichis a weaker condition than their incommensurability usedin Theorem 1), the matrix Φ ∞ in (36) becomes the identitymatrix and (35) reduces to E ∞ ( XX T ) = n / ∑ k = (cid:0) C k Γ C ∗ k + C k Γ C T k (cid:1) = n / ∑ k = (cid:104) V k V k (cid:105)(cid:34) W k Γ W ∗ k W k Γ W T k (cid:35)(cid:34) V ∗ k V T k (cid:35) . (41)6uch energy matrices R (cid:31) S n .The corresponding infinite-horizon average of a quadraticform of X is E ∞ ( X T Π X ) = ∑ n / k = (cid:0) V ∗ k Π V k W k Γ W ∗ k + V T k Π V k W k Γ W T k (cid:1) for any Π ∈ S n . Lemmas 2 and 3 can beused for computing quadratic cost functionals for QHOs,such as the performance criterion in the mean square op-timal CQF problem of Section 6. Furthermore, Lemma 3can be easily extended to the more general case R (cid:60) (cid:62) in (8). Example 1.
Consider a two-mode QHO of dimension n = Θ : = I ⊗ J , where ⊗ is the Kroneckerproduct, and J : = (cid:34) − (cid:35) (42)spans the space A . This corresponds to the system variablesconsisting of two pairs of conjugate position q k and mo-mentum − i ∂ q k operators (with an appropriately normalisedPlanck constant [42]), k = ,
2. Suppose the QHO has theenergy matrix R : = . . − . − . . . − . − . − . − . . . − . − . . . (cid:31) , so that the frequencies are ± . ± . τ ∗ = . Σ : = . − . − . . − . . . − . − . . . − . . − . − . . and satisfies the uncertainty relation constraint Σ + i Θ (cid:60) τ is depicted inFig. 1. These graphs show that, in the example being consid-ered, the interval 0 < τ < τ ∗ = . E ∞ ( XX T ) = . − . . . − . . . − . . . . − . . − . − . . . This matrix is calculated using the frequencies of the QHOin accordance with (41). (cid:78)
Consider a direct coupling of a quantum plant and a co-herent quantum observer which form a closed QHO whose
Fig. 1. The dependence of the real parts P jk of the discounted sec-ond moments of the system variables on the effective time horizon τ for the four-dimensional QHO of Example 1. The dashed linesrepresent the corresponding infinite-horizon averages as τ → + ∞ ,while the “ ◦ ”s indicate the initial values Re E ( X j ( ) X k ( )) . Hamiltonian H is given by H : = X T R X , X : = (cid:34) X ξ (cid:35) , X : = X ... X n , ξ : = ξ ... ξ ν , (43)where R ∈ S n + ν is the plant-observer energy matrix. Here, X , . . . , X n and ξ , . . . , ξ ν are the dynamic variables of theplant and the observer, respectively, with both dimensions n and ν being even. The plant and observer variables are time-varying self-adjoint operators on the tensor-product space H : = H ⊗ H , where H and H are initial complexseparable Hilbert spaces of the plant and the observer (whichcan be copies of a common Hilbert space). These quantumvariables are assumed to satisfy the CCRs with a block-diagonal CCR matrix Θ : [ X , X T ] = i Θ , Θ : = diag k = , ( Θ k ) , (44)where Θ ∈ A n and Θ ∈ A ν are nonsingular CCR matricesof the plant and the observer, respectively. For what follows,the plant-observer energy matrix R in (43) is partitioned as R : = (cid:34) K LL T M (cid:35) . (45)Here, K ∈ S n and M ∈ S ν are the energy matrices of the plantand the observer which specify their free Hamiltonians H : = X T KX and H : = ξ T M ξ . Also, L ∈ R n × ν is the plant-observer coupling matrix which parameterizes the interac-tion Hamiltonian H : = ( X T L ξ + ξ T L T X ) = Re ( X T L ξ ) ,where Re ( · ) applies to operators (and matrices of operators)so that Re N : = ( N + N ) consists of self-adjoint operators.Accordingly, the total Hamiltonian H in (43) is representableas H = H + H + H . In view of (43)–(45), the Heisenberg7ynamics of the composite system are governed by a linearODE ˙ X = i [ H , X ] = A X . (46)Here, in accordance with the partitioning of X in (43), thematrix A ∈ R ( n + ν ) × ( n + ν ) is split into appropriately dimen-sioned blocks as A : = Θ R = (cid:34) Θ K Θ L Θ L T Θ M (cid:35) = (cid:34) A BL β L T α (cid:35) , (47)with the ODE (46) being representable as a set of two ODEs˙ X = AX + B η , (48)˙ ξ = αξ + β Y , (49)where A : = Θ K , B : = Θ , (50) α : = Θ M , β : = Θ , (51) Y : = L T X , η : = L ξ . (52)The vector η drives the plant variables in (48), thus re-sembling the classical actuator signal. The observer vari-ables in (49) are driven by the plant variables through thevector Y which corresponds to the classical observationoutput from the plant. However, the quantum mechanicalnature of Y and η (which consist of time-varying self-adjoint operators on H ) makes them qualitatively differ-ent from the classical signals [2,25]. In view of the relation [ Y , Y T ] = L T [ X , X T ] L = iL T Θ L , following from (44) and(52), the outputs Y , . . . , Y ν do not commute with each other,in general, which makes them inaccessible to simultaneousmeasurement. Since the plant and the observer being con-sidered form a fully quantum system which does not involvemeasurements, Y is not an observation signal in the usualcontrol theoretic sense. In order to emphasize this distinctionfrom the classical case, the above described observers arereferred to as coherent (that is, measurement-free) quantumobservers [23,26,31,33,53,61]. In addition to the noncom-mutativity of the dynamic variables, specified by the CCRs(44), the quantum mechanical nature of the setting mani-fests itself in the fact that the “observation” and “actuation”channels in (52) depend on the same matrix L . This cou-pling between the ODEs (48) and (49) is closely related tothe Hamiltonian structure A ∈ Θ S n + ν of the matrix A in(47). Therefore, the “quantum information flow” from theplant to the observer through Y has a “back-action” effecton the plant dynamics through η . However, unlike the con-ventional meaning of this term in the context of quantummeasurements, the back-action considered here is caused bythe direct coupling of the observer which modifies the dy-namics of the plant.Assuming that the plant energy matrix K is fixed, the ma-trices L and M can be varied so as to achieve desired prop-erties for the plant-observer QHO under constraints on theplant-observer coupling. To this end, for a given effectivetime horizon τ >
0, the observer will be called τ -admissible if the matrix A in (47) satisfies τ < ( , ln r ( e A )) , (53)cf. (25) of Lemma 2. The corresponding pairs ( L , M ) form anopen subset of R n × ν × S ν which depends on τ . In applicationto the plant-observer system, the discussions of Section 3show that if the matrix R in (45) is positive definite (andhence, A has a purely imaginary spectrum), then such anobserver is τ -admissible for any τ >
0. The condition R (cid:31) K (cid:31) , M (cid:31) , (cid:107) Λ (cid:107) ∞ < , Λ : = K − / LM − / , (54)where the third inequality describes the contraction propertyof the matrix Λ whose largest singular value (cid:107) Λ (cid:107) ∞ quantifiesthe “smallness” of the coupling matrix L in comparison withthe energy matrices K and M . If the observer satisfies (54),then any system operator (with appropriate finite moments)in the plant-observer QHO lends itself to the discountedaveraging, described in Section 3, for any effective timehorizon τ >
0. Also note that the rescaling (cid:98) X : = √ KX , (cid:98) ξ : = √ M ξ (55)of the plant and observer variables leads to a QHO withappropriately transformed CCR matrices (cid:98) Θ : = √ K Θ √ K and (cid:98) Θ : = √ M Θ √ M , and the energy matrix (cid:98) R : = (cid:34) I n ΛΛ T I ν (cid:35) ,where Λ from (54) plays the role of the coupling matrix.For what follows, it is assumed that the initial plant andobserver variables have a block diagonal matrix of secondmoments: Σ : = Re E ( X X T0 ) = diag k = , ( Σ k ) , (56)where Σ k + i Θ k (cid:60) E X =
0, this corresponds to X and ξ being uncorrelated. A physical rationale for the absence ofinitial correlation is that the observer is prepared indepen-dently of the plant and then brought into interaction withthe latter at t =
0. If the plant and the observer remained un-coupled (which would correspond to the case L = E ( X ξ T ) =
0) and the corre-sponding matrices P : = Re E τ ( XX T ) and P : = Re E τ ( ξ ξ T ) would be unique solutions of independent ALEs: P = τ L ( A τ , Σ ) , A τ : = A − τ I n , (57) P = τ L ( α τ , Σ ) , α τ : = α − τ I ν , (58)where (22) is used, and both matrices A τ and α τ are assumedto be Hurwitz. In the general case of plant-observer coupling8 (cid:54) =
0, the matrix P : = (cid:34) P P P P (cid:35) : = Re E τ ( X X T ) , (59)which is split into blocks similarly to A in (47), coincideswith the controllability Gramian of the pair ( A τ , √ τ − Σ ) and satisfies an appropriate ALE: P = τ L ( A τ , Σ ) , (60)provided the observer is τ -admissible in the sense of (53).Here, Σ is the initial covariance condition from (56), and thematrix A τ : = A − τ I n + ν = (cid:34) A τ BL β L T α τ (cid:35) (61)is Hurwitz. As mentioned above, in the case L = P reduces to the block diagonal matrix P ∗ : = diag k = , ( P k ) (62)which is formed from the matrices P , P in (57) and (58). The back-action of the observer can be quantified by thedeviation of the covariance dynamics of the plant from thosewhich the plant would have if it were uncoupled from theobserver. In view of (59) and (62), we will describe thisdeviation in terms of bilateral bounds for P − P and, moregenerally, P − P ∗ . To this end, we will use the followingtechnical lemma whose proof is given here for completeness. Lemma 4
Suppose a matrix N ∈ (cid:34) N N N N (cid:35) ∈ S + n is splitinto blocks N jk ∈ R n × n . Then ± ( N + N ) (cid:52) wN + w N (63) for any w > . Furthermore, if N (cid:31) (in addition to N (cid:60) ),then ± ( N + N ) (cid:52) (cid:113) r ( N − N ) N . (64) Proof.
Positive semi-definiteness of the matrix N impliesthat 0 (cid:52) (cid:104) √ wI n ± √ w I n (cid:105) N (cid:34) √ wI n ± √ w I n (cid:35) = wN + w N ± ( N + N ) for any w >
0, which proves (63) (similar inequalitiesare used, for example, in the proof of [45, Lemma 3]). Theparameter w can be varied so as to “tighten up” the bound(63). More precisely, from the additional assumption N (cid:31)
0, it follows that wN + w N = √ N (cid:0) wI n + w N − / N N − / (cid:1) √ N (cid:52) (cid:0) w + w r ( N − N ) (cid:1) N . (65) The scalar coefficient on the right-hand side of this inequalityachieves its minimum valuemin w > (cid:0) w + w r ( N − N ) (cid:1) = (cid:113) r ( N − N ) (66)at w = (cid:113) r ( N − N ) , in which case, a combination of (63),(65) and (66) leads to (64). (cid:4) The following lemma will be used to give a more precisemeaning to the property that the observer output with rela-tively small mean square values has an appropriately weakeffect on the covariance dynamics of the plant.
Lemma 5
Suppose the directly coupled observer is θ -admissible, where ς : = w τ w + τ < τ < θ : = m τ m − τ (67) are related to the effective time horizon τ through auxiliaryparameters w > and m > τ . Then the matrix P from(59) satisfies − L (cid:0) A ς , w P + wB Re E τ ( ηη T ) B T (cid:1) (cid:52) P − P (cid:52) L (cid:0) A θ , m P + mB Re E τ ( ηη T ) B T (cid:1) , (68) where P is given by (57). Proof.
In view of the inequalities in (67), the θ -admissibilityof the observer ensures that all three matrices A ς , A τ and A θ are Hurwitz. Now, from the ODE (48), it follows that ( XX T ) (cid:5) = AXX T + XX T A T + B η X T + X η T B T . Applicationof the discounted averaging operator E τ to the latter ODEand the integration by parts on its left-hand side lead to τ ( P − Σ ) = A P + P A T + ϒ , and hence, A τ P + P A T τ + τ Σ + ϒ = A τ ( P − P ) + ( P − P ) A T τ + ϒ = . (69)Here, the term ϒ : = Re E τ ( B η X T + X η T B T ) (70)originates from the plant-observer coupling and plays therole of a perturbation to the ALE A τ P + P A T τ + τ Σ = N : = Re E τ ( ζ ζ T ) = (cid:34) P Re E τ ( X η T ) B T B Re E τ ( η X T ) B Re E τ ( ηη T ) B T (cid:35) (cid:60) ζ : = (cid:34) XB η (cid:35) , it follows that the matrix ϒ in (70) satisfies ϒ (cid:52) w P + wB Re E τ ( ηη T ) B T (cid:60) − ϒ (71)for any w >
0. Substitution of the second inequality from971) into (69) leads to0 (cid:60) A τ ( P − P ) + ( P − P ) A T τ − w P − wB Re E τ ( ηη T ) B T = A ς ( P − P ) + ( P − P ) A T ς − w P − wB Re E τ ( ηη T ) B T , (72)where use is also made of the relation τ + w = ς whichfollows from the definition of ς in (67) and implies that A τ − w I n = A ς in view of (57). The Lyapunov inequality(72) leads to the lower bound for P − P in (68). Theupper bound in (68) is established in a similar fashion bycombining the ALE (69) with the first inequality from (71),except that the parameter m : = w has to satisfy m > τ inorder to ensure that θ > A τ + m I n = A θ Hurwitz. (cid:4)
The parameters w and m in Lemma 5 can be varied in orderto tighten up the bounds (68), similarly to the proof of (64)of Lemma 4. Indeed, suppose the matrix B Re E τ ( ηη T ) B T issmall in comparison with P in terms of the dimensionlessquantity κ : = τ (cid:113) r ( P − B Re E τ ( ηη T ) B T ) , (73)provided P (cid:31)
0. The latter condition is fulfilled, for exam-ple, if Σ (cid:31) w = m = τκ in (67),the corresponding ς = τ + κ and θ = τ − κ become close to τ for small values of κ , and the bounds (68) behave asymp-totically as ± ( P − P ) (cid:45) κτ L ( A τ , P ) (cid:52) κ r ( P Σ − ) P . (74)The second inequality in (74) is obtained by applying(23) to (57). Therefore, Lemma 5 guarantees that the de-viation P − P is small in comparison with P (that is, r ( P P − − I n ) (cid:28)
1) and the back-action effect is neg-ligible, if the second moments of the observer output η are small enough in the sense that the parameter κ in (73)satisfies κ max ( , r ( P Σ − )) (cid:28) η and the observer variables ξ .We will therefore provide a more accurate bound for the de-viation P − P , which takes into account the whole plant-observer dynamics (48)–(52), including the fact that ξ isdriven by the plant output Y . The formulation of the follow-ing theorem employs auxiliary matrices D : = diag ( A τ ⊕ A τ , α τ ⊕ α τ ) , (75) D : = diag ( A τ ⊕ α τ , α τ ⊕ A τ ) , (76) E : = (cid:34) I n ⊗ ( BL ) ( BL ) ⊗ I n ( β L T ) ⊗ I ν I ν ⊗ ( β L T ) (cid:35) , (77) E : = (cid:34) I n ⊗ ( β L T ) ( BL ) ⊗ I ν ( β L T ) ⊗ I n I ν ⊗ ( BL ) (cid:35) , (78)where N ⊕ Q : = N ⊗ I + I ⊗ Q is the Kronecker sum of ma-trices. The matrices D and D are associated with the L -independent diagonal blocks of the matrix A τ from (61), while E and E depend linearly on the coupling matrix L and are associated with the L -dependent off-diagonal part of A τ . Theorem 6
Suppose the observer is τ -admissible, and bothmatrices A τ and α τ in (57) and (58) are also Hurwitz.Furthermore, let the plant-observer system have the block-diagonal initial covariance condition (56), and suppose thematrices ∆ : = D − E , ∆ : = D − E , (79) defined in terms of (75)–(78), satisfy the condition ε : = (cid:107) ∆ (cid:107) ∞ (cid:107) ∆ (cid:107) ∞ < . (80) Then the Frobenius norm of the deviation of the matrix P in (59) from its value P ∗ for uncoupled plant and observerin (62) admits upper bounds (cid:107) P − P (cid:107) (cid:54) ε − ε (cid:107) P ∗ (cid:107) , (81) (cid:107) P − P ∗ (cid:107) (cid:54) √ + (cid:107) ∆ (cid:107) ∞ − ε (cid:107) ∆ (cid:107) ∞ (cid:107) P ∗ (cid:107) . (82) Proof.
Despite the symmetry of the matrix P , we will use itsfull (rather than half-) vectorization vec ( P ) ∈ R ( n + ν ) [27].For brevity, the vectorization of a matrix will be written as (cid:126) ( · ) throughout the proof. The vector (cid:126) P can be obtained byappropriately permutating the entries of the vector (cid:126) P (cid:126) P (cid:126) P (cid:126) P .The latter satisfies the following vectorized form of the ALE(60) in view of (61): A τ ⊕ A τ I n ⊗ ( BL ) ( BL ) ⊗ I n I n ⊗ ( β L T ) A τ ⊕ α τ ( BL ) ⊗ I ν ( β L T ) ⊗ I n α τ ⊕ A τ I ν ⊗ ( BL ) ( β L T ) ⊗ I ν I ν ⊗ ( β L T ) α τ ⊕ α τ (cid:126) P (cid:126) P (cid:126) P (cid:126) P = − τ (cid:126) Σ (cid:126) Σ . (83)The sparsity of the right-hand side of (83) results from theblock-diagonal structure of the matrix Σ in (56) and splitsthe set of linear equations into the non-homogeneous andhomogeneous parts D (cid:34) (cid:126) P (cid:126) P (cid:35) + E (cid:34) (cid:126) P (cid:126) P (cid:35) = − τ (cid:34) (cid:126) Σ (cid:126) Σ (cid:35) , (84) D (cid:34) (cid:126) P (cid:126) P (cid:35) + E (cid:34) (cid:126) P (cid:126) P (cid:35) = , (85)where (75)–(78) are used. With the matrices A τ and α τ beingHurwitz, both D and D are nonsingular. Hence, by solving(85) for (cid:34) (cid:126) P (cid:126) P (cid:35) and substituting the solution into (84), it10ollows that (cid:34) (cid:126) P (cid:126) P (cid:35) = − ∆ (cid:34) (cid:126) P (cid:126) P (cid:35) , (86) (cid:34) (cid:126) P (cid:126) P (cid:35) = − τ ( D − E ∆ ) − (cid:34) (cid:126) Σ (cid:126) Σ (cid:35) = − τ ( I n + ν − ∆ ∆ ) − D − (cid:34) (cid:126) Σ (cid:126) Σ (cid:35) = ( I n + ν − ∆ ∆ ) − (cid:34) (cid:126) P (cid:126) P (cid:35) . (87)Here, use is made of (79) and (80) together with the vec-torized representations (cid:126) P = − τ ( A τ ⊕ A τ ) − (cid:126) Σ and (cid:126) P = − τ ( α τ ⊕ α τ ) − (cid:126) Σ for the solutions of the ALEs (57) and(58). Since the matrix ∆ : = ∆ ∆ is a contraction, with (cid:107) ∆ (cid:107) ∞ (cid:54) ε , application of the perturbation expansion for thematrix inverse [17] to (87) yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) (cid:126) P − (cid:126) P (cid:126) P − (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ ( I n + ν − ∆ ) − (cid:34) (cid:126) P (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ ∑ k = ∆ k (cid:34) (cid:126) P (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) + ∞ ∑ k = ε k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) (cid:126) P (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ε − ε | (cid:126) P ∗ | . (88)This implies (81) since | (cid:126) P ∗ | = (cid:113) ∑ k = (cid:107) P k (cid:107) = (cid:107) P ∗ (cid:107) inview of the preservation of the Frobenius norm under thevectorization. In order to prove (82), we note that the triangleinequality and (88) lead to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) (cid:126) P (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) (cid:126) P (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) (cid:126) P − (cid:126) P (cid:126) P − (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:107) P ∗ (cid:107) − ε , (89)where the identity (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) (cid:126) P (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:107) P ∗ (cid:107) is used again. A combi-nation of (89) with (86) implies that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) (cid:126) P (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:107) ∆ (cid:107) ∞ (cid:107) P ∗ (cid:107) − ε . (90)By using the orthogonal decomposition P − P ∗ = diag k = , ( P kk − P k ) + (cid:34) P P (cid:35) together with (88), (90),it follows that (cid:107) P − P ∗ (cid:107) = (cid:118)(cid:117)(cid:117)(cid:116)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) (cid:126) P − (cid:126) P (cid:126) P − (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) (cid:126) P (cid:126) P (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) √ ε + (cid:107) ∆ (cid:107) ∞ − ε (cid:107) P ∗ (cid:107) , which establishes (82) in view of (80). (cid:4) Since Theorem 6 employs the standard (rather thanweighted) Frobenius norm (cid:107) · (cid:107) , it would be physicallymore meaningful to apply the theorem to covariance dy-namics of the rescaled plant and observer variables (55).Alternatively, Theorem 6 can be reformulated in terms of an appropriately weighted version of the norm. In the lattercase, (cid:107) P − P ∗ (cid:107) is replaced with (cid:107) S ( P − P ∗ ) S (cid:107) , where S : = diag ( √ K , √ M ) . We have used the standard Frobeniusnorm in (81) and (82) merely for simplicity of formulation.There is a parallel between the proof of Theorem 6 and thearguments underlying the small-gain theorem (see, for ex-ample, [11] and references therein). Similar bounds for theobserver back-action can be obtained in the frequency do-main as outlined below. From (48), (49) and (52), it followsthat the Laplace transforms (cid:101) X and (cid:101) ξ of the plant and observervectors X and ξ , defined according to (32), are related by (cid:101) X ( s ) = F ( s )( X + BL (cid:101) ξ ( s )) , (91) (cid:101) ξ ( s ) = Φ ( s )( ξ + β L T (cid:101) X ( s )) , (92)see Fig. 2. Here, F and Φ are the plant and observer transfer F ( s ) Φ ( s ) BL (cid:54)(cid:54)(cid:63)(cid:63) (cid:110) + (cid:45)(cid:45) ξ (cid:101) ξ ( s ) β L T (cid:27) (cid:110) + (cid:27) X (cid:101) X ( s ) Fig. 2. A block diagram of Eqs. (91) and (92), with the initial values X and ξ shown as fictitious external inputs. A small-gain-theoremargument applies when the coupling matrix L is relatively small. functions, which are given by F ( s ) : = ( sI n − A ) − , Φ ( s ) : = ( sI ν − α ) − (93)in accordance with (31) and do not depend on the cou-pling matrix L . It follows from (91)–(93) that the Laplacetransform of the combined vector X of the plant and ob-server variables in (43) is related to its initial value X by (cid:102) X ( s ) = (cid:34) (cid:101) X ( s ) (cid:101) ξ ( s ) (cid:35) = G ( s ) X through the transfer function G ( s ) : = (cid:34) I n − F ( s ) BL − Φ ( s ) β L T I ν (cid:35) − (cid:34) F ( s ) Φ ( s ) (cid:35) . (94)By applying (29) to the plant-observer system, the matrix P in (59) is represented as P = πτ Im (cid:90) Re s = τ G ( s )( Σ + i Θ ) G ( s ) ∗ d s . (95)The function G in (94) differs from diag ( F , Φ ) by the factor (cid:34) I n − T ( s ) − T ( s ) I ν (cid:35) − = (cid:34) ( I n − T ( s ) T ( s )) − ( I n − T ( s ) T ( s )) − T ( s )( I ν − T ( s ) T ( s )) − T ( s ) ( I ν − T ( s ) T ( s )) − (cid:35) which is close to I n + ν for the relevant values of s ∈ C (withRe s = τ ), provided the coupling matrix L is small in the11ense of the quantities γ k : = sup ω ∈ R (cid:107) T k ( τ + i ω ) (cid:107) ∞ , k = , . (96)The latter are “discounted” versions of the H ∞ Hardyspace norm for the transfer functions T ( s ) : = F ( s ) BL and T ( s ) : = Φ ( s ) β L T which depend linearly on L . There-fore, the frequency-domain representation (95) can beused together with the parameters (96) in order to ob-tain bounds for the deviation of P from the matrix P ∗ = πτ Im (cid:82) Re s = τ diag ( F ( s ) Γ F ( s ) ∗ , Φ ( s ) Γ Φ ( s ) ∗ ) d s under the small-gain condition γ γ < Γ k : = Σ k + i Θ k , with k = ,
2, are the initial second mo-ment matrices for the plant and observer variables, which,in accordance with (56), form the corresponding matrix forthe closed-loop system: Γ : = E ( X X T0 ) = diag k = , ( Γ k ) . For what follows, let the plant energy matrix K be fixed andsatisfy K (cid:31)
0. The latter is sufficient (but not necessary) forthe set of τ -admissible observers to be nonempty for anygiven τ >
0. In particular, this set contains observers with L = M (cid:31) R = diag ( K , M ) (cid:31) Z : = E τ Z −→ min (97)of minimizing a quadratic cost functional over the plant-observer coupling matrix L and the observer energy matrix M subject to the constraint (53). Here, τ > Z : = E T E + λ η T Π η = X T C T C X . (98)The latter is a time-varying self-adjoint operator on the plant-observer space H which is defined in terms of the vectors X , η from (43), (52), and E : = S X − S ξ = S X , S : = (cid:104) S − S (cid:105) , C : = (cid:34) S − S √ λ Π L (cid:35) . (99)Here, S ∈ R p × n , S ∈ R p × ν and Π ∈ S n are given matrices,with Π (cid:31)
0, which, together with a given scalar parameter λ >
0, determine the matrix C ∈ R ( p + n ) × ( n + ν ) (with the firstblock-row S ∈ R p × ( n + ν ) ) and its dependence on the cou-pling matrix L . The matrix S specifies linear combinationsof the plant variables of interest which are to be approx-imated by given linear functions of the observer variablesspecified by the matrix S . Accordingly, the vector E in(99) (consisting of p time-varying self-adjoint operators on H ) is interpreted as an estimation error . In addition to thediscounted mean square E τ ( E T E ) of the estimation error,the cost functional Z in (97) involves a quadratic penalty E τ ( η T Π η ) for the observer back-action on the covariancedynamics of the plant (see Lemma 5), with λ being the rel-ative weight of this penalty in Z . In fact, Z is organised as the Lagrange function for a related CQF problem of mini-mizing the discounted mean square of the estimation errorsubject to an additional weighted mean square constraint onthe plant-observer coupling: E τ ( E T E ) −→ min , E τ ( η T Π η ) (cid:54) r . (100)In this formulation, λ plays the role of a Lagrange multi-plier which is found so as to make the solution of (97) sat-urate the constraint in (100) for a given threshold r >
0. Ina particular case S =
0, the CQF problem (97)–(99) is aquantum mechanical analogue of the LQR problem [2,25]in view of the analogy between the observer output η andclassical actuation signals discussed in Section 4. The pres-ence of the quantum expectation of a nonlinear function ofsystem variables in (97) and the optimization requirementmake this setting different from the time-averaged approachof [38,40].Substitution of (98) into (97) allows the cost functional tobe expressed in terms of the matrix P from (59) as Z = (cid:104) C T C , E τ ( X X T ) (cid:105) = (cid:104) C T C , P (cid:105) , (101)where (cid:104)· , ·(cid:105) is the Frobenius inner product of matrices. Underthe assumptions of Theorem 6, a combination of (101) withthe Cauchy-Bunyakovsky-Schwarz inequality and the bound(82) leads to | E τ ( E T E ) − (cid:104) S T S , P ∗ (cid:105)| = |(cid:104) S T S , P − P ∗ (cid:105)| (cid:54) (cid:107) S T S (cid:107) (cid:107) P − P ∗ (cid:107) (cid:54) (cid:107) S T S (cid:107) √ + (cid:107) ∆ (cid:107) ∞ − ε (cid:107) ∆ (cid:107) ∞ (cid:107) P ∗ (cid:107) , (102)which relates the discounted mean square of the estima-tion error with the plant-observer coupling strength quan-tified by (cid:107) ∆ (cid:107) ∞ , (cid:107) ∆ (cid:107) ∞ and ε from (79) and (80). Here, S is the first block-row of the matrix C in (99), so that (cid:107) S T S (cid:107) = (cid:107) SS T (cid:107) = (cid:113) Tr (( ∑ k = S k S T k ) ) , and (cid:104) S T S , P ∗ (cid:105) = ∑ k = Tr ( S k P k S T k ) in view of the block-diagonal structure ofthe matrix P ∗ in (62). Therefore, the inequality (102) im-plies that E τ ( E T E ) (cid:62) ∑ k = Tr ( S k P k S T k ) − (cid:118)(cid:117)(cid:117)(cid:116) Tr (cid:16)(cid:16) ∑ k = S k S T k (cid:17) (cid:17) √ + (cid:107) ∆ (cid:107) ∞ − ε (cid:107) ∆ (cid:107) ∞ (cid:107) P ∗ (cid:107) , (103)which becomes an equality if L =
0. The right-hand side of(103) provides a lower bound for the mean square of theestimation error. This bound depends on the coupling ma-trix L only through (cid:107) ∆ (cid:107) ∞ , (cid:107) ∆ (cid:107) ∞ , ε and shows that L hasto be sufficiently large in order to make E τ ( E T E ) smallerthan ∑ k = Tr ( S k P k S T k ) by a given amount. At the same time,the plant-observer coupling should be weak enough to avoidsevere back-action of the observer on the plant. Therefore,the parameter λ in the CQF problem (97)–(99) quantifies a12ompromise between these conflicting requirements (of min-imizing the estimation error and reducing the back-action). The following theorem provides first-order necessary con-ditions of optimality for the CQF problem (97)–(99). Theirformulation employs the Hankelian E : = (cid:34) E E E E (cid:35) : = QP , (104)associated with the matrix P from (59) and the observabilityGramian Q of ( A τ , C ) which is a unique solution of thecorresponding ALE: Q : = (cid:34) Q Q Q Q (cid:35) = L ( A T τ , C T C ) . (105)The matrices E and Q are split into appropriately dimen-sioned blocks ( · ) jk similarly to the matrix P in (59), with ( · ) j • the j th block-row and ( · ) • k the k th block-column ofthe matrices. Theorem 7
Suppose the plant energy matrix satisfies K (cid:31) ,and the directly coupled observer is τ -admissible in the senseof (53). Then the observer is a stationary point of the CQFproblem (97)–(99) if and only if the Hankelian E in (104)and the controllability Gramian P in (59) satisfy Θ E − E T21 Θ = λ Π L P , (106) Θ E − E T22 Θ = . (107) Proof.
By using (60) and the duality L ( A τ , · ) † = L ( A T τ , · ) ,it follows that the cost Z in (101) is representable in termsof the observability Gramian Q from (105) as Z = τ (cid:10) C T C , L ( A τ , Σ ) (cid:11) = τ (cid:10) L ( A T τ , C T C ) , Σ (cid:11) = τ (cid:104) Q , Σ (cid:105) . (108)Here, the adjoint ( · ) † of linear operators on matrices is in thesense of the Frobenius inner product. With the matrix A τ in(61) being Hurwitz due to the τ -admissibility constraint (53),the representation (108) shows that Z inherits a smoothdependence on L and M from Q . The latter is a compositefunction ( L , M ) (cid:55)→ ( A , C ) (cid:55)→ Q whose first variation is δ Q = L ( A T τ , ( δ A ) T Q + Q δ A + ( δ C ) T C + C T δ C ) , (109)where use is made of the ALE in (105), and the first varia-tions of the matrices A in (47) and C in (99) with respectto L and M are δ A = Θ (cid:34) δ L δ L T δ M (cid:35) , δ C = (cid:34) √ λ Π δ L (cid:35) . (110)By combining the duality argument above with (109) and(110), it follows that the first variation of Z in (108) can be computed as δ Z = τ (cid:10) L ( A T τ , ( δ A ) T Q + Q δ A + ( δ C ) T C + C T δ C ) , Σ (cid:11) = (cid:10) ( δ A ) T Q + Q δ A + ( δ C ) T C + C T δ C , P (cid:11) = (cid:104) E , δ A (cid:105) + (cid:104) C P , δ C (cid:105) = − (cid:42) Θ E , (cid:34) δ L δ L T δ M (cid:35)(cid:43) + (cid:42) C P , (cid:34) √ λ Π δ L (cid:35)(cid:43) = − (cid:42) S ( Θ E ) , (cid:34) δ L δ L T δ M (cid:35)(cid:43) + (cid:68) ( C P ) , √ λ Π δ L (cid:69) = − (cid:104) S ( Θ E ) , δ L (cid:105) − (cid:104) S ( Θ E ) , δ M (cid:105) + (cid:104)√ λ Π L P , √ λ Π δ L (cid:105) = (cid:104) λ Π L P − S ( Θ E ) , δ L (cid:105) − (cid:104) S ( Θ E ) , δ M (cid:105) (111)(similar calculations can be found, for example, in [52]).Here, S ( N ) : = ( N + N T ) denotes the symmetrizer of ma-trices, so that S ( Θ E ) = ( Θ E − E T Θ ) = (cid:34) Θ E − E T11 Θ Θ E − E T21 Θ Θ E − E T12 Θ Θ E − E T22 Θ (cid:35) . (112)A combination of (111) with (112) leads to the partialFrechet derivatives of Z on the corresponding Hilbertspaces of matrices R n × ν and S ν : ∂ L Z = ( λ Π L P − S ( Θ E ) )= ( λ Π L P − ( Θ E − E T21 Θ )) , (113) ∂ M Z = − S ( Θ E ) = − ( Θ E − E T22 Θ ) . (114)By equating the Frechet derivatives (113) and (114) to zero,it follows that the stationarity of Z with respect to L and M is equivalent to (106) and (107). (cid:4) The relation (112) implies that the fulfillment of the first-order optimality conditions (106) and (107) for the observeris equivalent to the existence of a matrix N ∈ S n such that Θ E − E T Θ = (cid:34) N λ Π L P λ P L T Π (cid:35) . (115)Here, the zero block corresponds to (107), which meansthat the matrix E is skew-Hamiltonian in the sense of thesymplectic structure specified by Θ − , that is, E ∈ Θ − A ν .A quantum probabilistic interpretation of the optimality con-ditions (106) and (107) is that, for any such observer, theprocess ϑ , given by ϑ : = (cid:34) ϑ ϑ (cid:35) : = Θ QX , ϑ j : = Θ j Q j • X , j = , , (116)and consisting of n + ν self-adjoint operators (which are spe-cial linear combinations of the plant and observer variables),13atisfies the covariance relations E τ ( ϑ ξ T + X ϑ T2 ) = Θ Q E τ ( X ξ T ) − E τ ( X X T ) Q • Θ = Θ Q (cid:34) P P + i Θ (cid:35) − ( P + i Θ ) Q • Θ = Θ E • − E T2 • Θ = (cid:34) λ Π L P (cid:35) = (cid:34) λ Π Re E τ ( ηξ T ) (cid:35) . (117)Here, use is made of the identities E jk = Q j • P • k and E T jk = P k • Q • j , which follow from (104) and the symmetry of theGramians P and Q in (59) and (105). In particular, (117)implies that ϑ in (116) and ξ are uncorrelated in the sensethat E τ ( ϑ ξ T + ξ ϑ T2 ) = . (118)This is a quantum counterpart of the corresponding propertyfor the state estimation error and the state estimate in theclassical Kalman filter [1].If P (cid:31)
0, then, in view of the assumption Π (cid:31)
0, (106)implies that the optimal coupling matrix is representable as L = λ Π − ( Θ E − E T21 Θ ) P − . (119)In order to close the ALEs (60) and (105), the relation (119)needs to be complemented with an appropriate equation forthe optimal observer matrix M . The latter step is less straight-forward and will be considered in the next section. For what follows, we associate with the Gramians P and Q from (59) and (105) the matrices P : = P Θ − , Q : = Θ Q (120)belonging to the same subspace Θ S n + ν of Hamiltonian ma-trices as A in (47). Here, the property P ∈ Θ S n + ν followsfrom Θ − P Θ − ∈ S n + ν . The linear space Θ S n + ν , equippedwith the commutator [ · , · ] , is a Lie algebra [12,34,49], interms of which the ALEs and the optimality conditions abovewill be reformulated by the following lemma. Its formula-tion employs the Hamiltonian matrix D : = [ Q , P ] = Θ QP Θ − − PQ = ( Θ E − E T Θ ) Θ − , (121)which (for any τ -admissible observer) is related to the left-hand side of (115) due to (104), (120) and the symmetry ofthe Gramians P , Q . Lemma 8
The ALEs (60), (105) and the optimality condi-tions (106), (107) for the CQF problem (97)–(99) are rep-resentable in a Lie-algebraic form through the Hamiltonianmatrices P, Q from (120): [ A , P ] = τ ( P − ΣΘ − ) , (122) [ A , Q ] = Θ C T C − τ Q , (123) D = λ Π LP , (124) D = , (125) where D and D are the corresponding blocks of thematrix D ∈ Θ S n + ν in (121). Proof.
The Hamiltonian structure of the matrix A in (47)implies that A T = − Θ − A Θ , and hence, A τ P + PA T τ = A P + PA T − τ P = A P − P Θ − A Θ − τ P = (cid:0) [ A , P ] − τ P (cid:1) Θ , (126) A T τ Q + QA τ = A T Q + QA − τ Q = − Θ − A Θ Q + QA − τ Q = − Θ − (cid:0) [ A , Q ] + τ Q (cid:1) , (127)where use is also made of (61) and (120). Substitution of(126) and (127) into the ALEs (60), (105) leads to theirLie-algebraic representations (122), (123). Furthermore,by substituting (115) into (121), considering the secondblock-column D • = (cid:34) λ Π L P (cid:35) Θ − and using the relation P Θ − = P , it follows that the optimality conditions(106) and (107) admit the Lie-algebraic representations(124) and (125). (cid:4) The solutions of (122) and (123) admit the representation P = ( I − τ ad A ) − ( ΣΘ − ) , (128) Q = τ ( I + τ ad A ) − ( Θ C T C ) , (129)where I is the identity operator on the space Θ S n + ν . Here,the resolvents ( I ± τ ad A ) − are well-defined since the τ -admissibility (53) implies that the spectrum of the linearoperator ad A on Θ S n + ν is contained in the strip { z ∈ C : | Re z | < τ } . Lemma 9
The optimal coupling matrix L in (119) can beexpressed in terms of the matrices P and Q from (120) asL = λ Π − D P − , (130) provided P (cid:31) , where the matrix D is given by (121).Furthermore, the optimal energy matrix M of the observersatisfies (cid:16) τ [ ΣΘ − , Q ] + [ Θ C T C , P ] (cid:17) + D Θ L − Θ KD + D Θ M = . (131) Proof.
The representation (130) follows directly from thefirst optimality condition (124) under the assumption P (cid:31)
0. In order to establish (131), we note that the left-handsides of (122)–(125) involve pairwise commutators of theHamiltonian matrices A , P , Q ∈ Θ S n + ν . Application of theJacobi identity [49] and the antisymmetry of the commutatorleads to the relations0 = [[ P , A ] , Q ] + [[ A , Q ] , P ] + [[ Q , P ] , A ]= τ [ ΣΘ − − P , Q ] + (cid:2) Θ C T C − τ Q , P (cid:3) + [ D , A ]= τ [ ΣΘ − , Q ] + [ Θ C T C , P ] + [ D , A ] (132)14or any τ -admissible observer, where use is made of (121)(here, neither of the optimality conditions (124) and (125)has been used). By substituting the matrix A from (47) intothe right-hand side of (132) and considering the ( · ) blockof the resulting Hamiltonian matrix, it follows that τ [ ΣΘ − , Q ] + [ Θ C T C , P ] + (cid:0) D Θ L + D Θ M − Θ ( KD + LD ) (cid:1) = . (133)Now, the second optimality condition (125) makes the cor-responding term in (133) vanish, thus leading to (131). (cid:4) As can be seen from the proof of Lemma 9, the relation(131) holds for any τ -admissible stationary point of the CQFproblem regardless of the assumption P (cid:31)
0. Furthermore,(131) is a linear equation with respect to M . This allows theoptimal observer energy matrix M to be expressed in termsof P , Q from (120) in the case of equal plant and observerdimensions n = ν . In this case, the observer will be called nondegenerate if the matrices P and D from (120) and (121)satisfy P (cid:31) , det D (cid:54) = . (134)The above results lead to the following necessary conditionsof optimality for such observers. Theorem 10
Suppose the plant and observer dimensionsare equal: n = ν . Then for any nondegenerate observer,which is a stationary point of the CQF problem (97)–(99)under the assumptions of Theorem 7, the coupling and en-ergy matrices are related by (130) andM = Θ − D − (cid:0) Θ KD − D Θ L − (cid:0) τ [ ΣΘ − , Q ] + [ Θ C T C , P ] (cid:1)(cid:1) (135) to the matrices P and Q from (120) satisfying the ALEs (122)and (123). Proof.
The first of the conditions (134) makes the represen-tation (130) applicable, which leads to a nonsingular cou-pling matrix L in view of the second condition in (134). Thelatter allows (131) to be uniquely solved for the observerenergy matrix M in the form (135). (cid:4) The first line of (135) is organised as a similarity transforma-tion which would relate the Hamiltonian matrices Θ K and Θ M if there were no additional terms on the right-hand sideof the equation. In that case, the transformation matrix D in (135) would preserve the Hamiltonian structure if it weresymplectic in the generalized sense that D Θ D T12 = Θ .In combination with the ALEs (60) and (105) (or their Lie-algebraic form (120)–(123), the relations (130) and (135) ofLemma 9 and Theorem 10 provide a set of algebraic equa-tions for finding the matrices L and M of a nondegenerateobserver among stationary points in the CQF problem (97)–(99). In view of the complicated structure of the equations of Sec-tions 7 and 8 for an optimal observer, consider a suboptimalsolution of the CQF problem in a special class of observerswhich lead to autonomous dynamics of the estimation error E in (99). More precisely, suppose the observer is such that S A = (cid:99) A S (136)for some (cid:99) A ∈ R p × p , where the matrix S is given by (99). Incombination with (46), the relation (136) leads to the ODE˙ E = S ˙ X = S A X = (cid:99) A S X = (cid:99) A E . (137)These autonomous dynamics preserve the CCRs for the es-timation error: [ E , E T ] = i (cid:98) Θ , (cid:98) Θ : = S Θ S T = ∑ k = S k Θ k S T k . (138)Indeed, from (136), (138) and the Hamiltonian property A ∈ Θ S n + ν , it follows that (cid:99) A (cid:98) Θ + (cid:98) Θ (cid:99) A T = (cid:99) A S Θ S T + S Θ S T (cid:99) A T = S ( A Θ + Θ A T ) S T = . (139)Therefore, if the CCR matrix (cid:98) Θ ∈ A p in (138) is nonsingular,then (139) implies that (cid:99) A is Hamiltonian in the sense that (cid:99) A ∈ (cid:98) Θ S p .Now, let the plant and the observer have equal dimensions n = ν and identical CCR matrices Θ : = Θ = Θ , (140)with Θ ∈ A n and det Θ (cid:54) =
0. Also, suppose the estimationerror E in (99) has the same dimension p = n and is specifiedby equal nonsingular matrices S : = S = S , (141)with S ∈ R n × n and det S (cid:54) =
0. Then the process E reducesto E = S ( X − ξ ) , (142)and its CCR matrix in (138) is nonsingular: (cid:98) Θ = S Θ S T0 . (143)Since E T E = ( X − ξ ) T S T0 S ( X − ξ ) in view of (142), thematrix S T0 S (cid:31) Lemma 11
Under the conditions (140) and (141), the esti-mation error (142) acquires the autonomous dynamics (137)due to (136) for some matrix (cid:99) A ∈ R n × n if and only if the bserver has the same energy matrix as the plant and a sym-metric coupling matrix:K = M , L = L T . (144) For any such observer, the matrix (cid:99) A is found uniquely as (cid:99) A = (cid:98) Θ (cid:98) R , (145) where (cid:98) Θ is the CCR matrix of the estimation error in (143),and (cid:98) R : = S − T0 ( K − L ) S − (146) is a real symmetric matrix of order n. Proof.
A combination of (47) with (140) and (141) leads to S A = S Θ (cid:104) K − L T L − M (cid:105) , (cid:99) A S = (cid:99) A S (cid:104) I n − I n (cid:105) . (147)Therefore, since det S (cid:54) =
0, the fulfillment of (136) for somematrix (cid:99) A ∈ R n × n is equivalent to K − L T = M − L , that is, M − K = L − L T . (148)Since the left-hand side of (148) is a symmetric matrix, whileits right-hand side is antisymmetric, and only the zero matrixhas these properties simultaneously ( S n (cid:84) A n = { } ), then(148) holds if and only if L and M satisfy (144). In this case,(136), (143) and (147) imply that (cid:99) A = S Θ ( K − L ) S − = (cid:98) Θ S − T0 ( K − L ) S − , which leads to (145), with (cid:98) R given by(146). (cid:4) The observer, described in Lemma 11, replicates the quan-tum plant, except that it is endowed with a different initialspace and, in general, different initial covariance conditionsin (56). The structure (144) of such observers does not de-pend on particular matrices Θ and S . In view of (137) and(145), the entries of the estimation error E in (142) evolvein time as system variables of a QHO with the CCR ma-trix (cid:98) Θ in (143) and the energy matrix (cid:98) R in (146). Withoutadditional constraints on the coupling matrix L (apart fromits symmetry in (144)), (cid:98) R can be ascribed any given valuein S n by an appropriate choice of L . However, large valuesof L are penalized by the second term of the cost functionalin (98). A solution of the CQF problem (97) in this class ofobservers is as follows. Theorem 12
In the framework of Lemma 11 under the con-ditions (140), (141) and P (cid:31) , an optimal coupling ma-trix L ∈ S n for the observer with autonomous estimation er-ror dynamics satisfiesL = − λ Π − L ( P Π − , S ( S ( Θ E ) )) Π − . (149) Proof.
In view of (144), the observer energy matrix M = K remains fixed, and, due to the symmetry of L , the first vari-ation (111) of the cost functional in the proof of Theorem 7 reduces to δ Z = (cid:104) S ( λ Π L P − S ( Θ E ) ) , δ L (cid:105) . Hence, ∂ L Z = S ( λ Π L P − S ( Θ E ) )= λ ( P L Π + Π L P ) − S ( S ( Θ E ) )= λ ( P Π − (cid:101) L + (cid:101) L Π − P ) − S ( S ( Θ E ) ) , (150)where (cid:101) L : = Π L Π (151)inherits its symmetry from L and Π . From (150), it followsthat ∂ L Z = (cid:101) L being a unique solution ofan appropriate ALE: (cid:101) L = − λ L ( P Π − , S ( S ( Θ E ) )) , (152)where P Π − is isospectral to Π − / P Π − / (cid:31)
0. Acombination of (151) with (152) leads to (149). (cid:4)
The right-hand side of the equation (149) is a nonlinearcomposite function of the coupling matrix L and a scalarparameter µ : = λ > L µ = µ f ( µ , L µ ) . (154)The computation of the function f involves the solution ofthe ALEs (60) and (105) for the Gramians P and Q withthe matrix A = (cid:34) Θ K Θ L Θ L Θ K (cid:35) , (155)followed by computing the Hankelian E in (104) and solvingthe ALE (152). The parameter µ in (153) enters f onlythrough the matrix C T C = S T S + (cid:34) µ L Π L (cid:35) (156)in the ALE (105). The smallness of µ corresponds to largevalues of λ (that is, high penalization of the observer back-action on the plant). For all sufficiently small µ > L ∈ S n , the function f is Frechet differentiable, and thissmoothness is inherited by L µ in (154). The differentiationof (154) with respect to µ (as fictitious time) leads to theODE ∂ µ L µ = ( I − µ∂ L f ) − ( f + µ∂ µ f ) , (157)with the initial condition L =
0, where I is the identityoperator on the space S n , and ∂ L f is the appropriate par-tial Frechet derivative of f . The initial-value problem (157)describes a homotopy method for numerical solution of theCQF problem, similar to [28] (see also, [50]). The right-handside of (157) is well-defined for all ( µ , L ) in a small neigh-bourhood of ( , ) . Its computation can be implemented byusing the vectorised representations of the Frechet deriva-tives of solutions of ALEs [47,52] in application to the ALEs(60), (105) (or their Lie-algebraic forms (128), (129)) and(152). The details of these calculations are tedious and omit-ted for brevity. The weak-coupling (or high-penalization)asymptotic behaviour of the matrix L µ is described below.16 heorem 13 Suppose the uncoupled observer has a posi-tive definite matrix P in (58). Then, for large values of theparameter λ in (98), the optimal coupling matrix in Theo-rem 12 satisfies the asymptotic relationL µ ∼ µ L (cid:48) , as µ → + , (158) where the matrix L (cid:48) ∈ S n is a unique solution of the ALEL (cid:48) = Π − L (cid:0) P Π − , Θ Q ( P + P ) − ( P + P ) Q Θ (cid:1) Π − . (159) Here, P and P are the second-moment matrices (57) and(58) for the uncoupled plant and observer variables given byP k = τ L ( A τ , Σ k ) , k = , , (160) with a common matrixA τ : = A − τ I n , A = Θ K . (161) Also, Q : = S T0 (cid:98) Q S (162) in (159) is associated with a unique solution (cid:98) Q of the ALE (cid:98) Q : = L ( (cid:99) A T τ , I n ) , (cid:99) A τ : = (cid:99) A − τ I n , (cid:99) A = S Θ KS − . (163) Proof.
From the representation (154) of (149) (or from(157)), it follows that (158) holds with L (cid:48) : = ∂ µ L µ (cid:12)(cid:12) µ = = f ( , ) , (164)where we have also used the initial condition L =
0. Here, f ( , ) = − Π − L ( P Π − , S ( S ( Θ E ) )) Π − (165)is associated with the uncoupled plant and observer, in whichcase they have the block-diagonal controllability Gramian in(62), where the matrices P and P are given by (160), (161)since the matrix (155) reduces to A = I ⊗ ( Θ K ) with apurely imaginary spectrum due to K (cid:31)
0. In the limit of un-coupled plant and observer, µ L µ Π L µ ∼ µ f ( , ) Π f ( , ) → µ → + , whereby (156) leads to C T C = S T S at µ = Q takesthe form A T τ Q + QA τ + S T S = . (166)The property (136) of the observers under consideration im-plies that S A τ = S A − τ S = (cid:99) A S − τ S = (cid:99) A τ S and hence,(166) admits a lower-rank solution Q = S T (cid:98) Q S . (167)Indeed, its substitution into the left-hand side of (166) yields A T τ Q + QA τ + S T S = S T ( (cid:99) A T τ (cid:98) Q + (cid:98) Q (cid:99) A τ + I n ) S . Therefore,(163) makes (167) a unique solution of the ALE (166), sincethe matrix (cid:99) A is isospectral to 2 Θ K with a purely imaginaryspectrum (so that (cid:99) A τ is Hurwitz). In view of S = (cid:104) − (cid:105) ⊗ S , the Hankelian takes the form E = S T (cid:98) Q S diag k = , ( P k ) = (cid:34) Q P − Q P − Q P Q P (cid:35) , with the matrix Q given by (162), andhence, S ( S ( Θ E ) ) = (( P + P ) Q Θ − Θ Q ( P + P )) . (168)Substitution of (168) into (165) and (164) leads to (159). (cid:4) Example 2.
Let the plant and observer be one-mode QHOs( n = ν = Θ : = J (correspondingto the position-momentum pair, with J given by (42), and apositive definite energy matrix K : = (cid:34) . − . − . . (cid:35) . The frequencies of such a QHO are ± . τ ∗ = . Σ = (cid:34) . − . − . . (cid:35) , Σ = (cid:34) . . . . (cid:35) and satisfy the uncertainty relation constraints Σ k + i Θ (cid:60) τ = . (cid:29) τ ∗ , the second-moment matrices of the uncoupled plant andobserver variables in (57), (58) are P = (cid:34) . . . . (cid:35) , P = (cid:34) . . . . (cid:35) . For the CQF problem (97), the observer back-action penaltymatrix Π in (98) and the weighting matrix S in the estima-tion error (142) are given by Π = (cid:34) . . . . (cid:35) , S = (cid:34) − . . . . (cid:35) . The mean square of the estimation error for the uncoupledobserver is Tr ( S ( P + P ) S T0 ) = . E τ ( E T E ) for the optimal observer in the CQF problem(97) (subject to the autonomous estimation error dynamics)is shown in Fig. 3 for a range of values of the parameter µ in (153). This is a monotonically decreasing function of µ , whose computation (along with the optimal observers)was carried out using the homotopy method starting fromthe uncoupled observer (at µ = λ E τ ( η T Π η ) is shown in Fig. 4. The entries ofthe corresponding optimal coupling matrix L µ are presentedin Fig. 5. The calculation of the matrix L (cid:48) , which specifiestheir asymptotic behaviour as µ → + according to Theo-rem 13, yielded L (cid:48) = (cid:34) − . − . − . . (cid:35) . ig. 3. The dependence of the mean square value E τ ( E T E ) of theestimation error on the parameter µ in Example 2.Fig. 4. The dependence of the back-action penalty term λ E τ ( η T Π η ) on the parameter µ in Example 2.Fig. 5. The dependence of the entries of the coupling matrix L µ on the parameter µ in Example 2. For the range 0 (cid:54) µ (cid:54) λ (cid:62) . (cid:34) K L µ L µ K (cid:35) remained positive definite, so thatthe system variables retained oscillatory behaviour, whichjustifies the discounted averaging approach. (cid:78)
10 Conclusion
We have considered the computation of discounted averageswith exponentially decaying weights for moments of systemvariables for QHOs, including the mean square functionals,both in the state space and frequency domain. For a quantumplant and a quantum observer in the form of directly cou-pled QHOs, we have obtained small-gain-theorem boundsfor the back-action of the observer on the covariance dy-namics of the plant in terms of the plant-observer coupling.We have considered a CQF problem of minimizing the dis-counted mean square value of the estimation error togetherwith a penalty on the observer back-action. First-order nec-essary conditions of optimality have been obtained for thisproblem in the form of a set of algebraic matrix equationsinvolving two coupled ALEs. We have applied Lie-algebraictechniques to these equations and discussed a solution ofthe CQF problem in the case of autonomous estimation er-ror dynamics, including the homotopy method for its imple-mentation. These results have been illustrated by numericalexperiments.
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