Characterization and Moment Stability Analysis of Quasilinear ...
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Characterization and Moment Stability Analysis of Quasilinear Quantum Stochastic Systems with Quadratic Coupling to External Fields∗ Igor G. Vladimirov† ,
Abstract— The paper is concerned with open quantum systems whose Heisenberg dynamics are described by quantum stochastic differential equations driven by external boson fields. The system-field coupling operators are assumed to be quadratic polynomials of the system variables, with the latter satisfying canonical commutation relations. In combination with a cubic system Hamiltonian, this leads to quasilinear quantum stochastic systems which extend the class of linear quantum systems and yet retain algebraic closedness in the evolution of mixed moments of system variables up to any order. Although such a system is nonlinear and its quantum state is no longer Gaussian, the dynamics of the moments are amenable to exact analysis, including the computation of their steady-state values. A generalized criterion is outlined for quadratic stability of the quasilinear systems.
I. I NTRODUCTION Open quantum systems, driven by external boson fields, are usually described in the Heisenberg picture by quantum stochastic differential equations (QSDEs) [14]. This class of dynamical systems, whose variables are noncommutative operators on a Hilbert space, is a common source of models in quantum optics [2] and quantum control [1], [5], [6], [21]. Central to this approach is a Markovian description of interaction between the quantum-mechanical system and its environment, whose “energetics” is specified by the system Hamiltonian and system-field coupling operators. The case where the Hamiltonian is quadratic and the coupling operators are linear with respect to the system variables, and the latter satisfy canonical commutation relations (CCRs) [12], corresponds to an open quantum harmonic oscillator. Its importance for linear quantum stochastic control [1], [13], [16], [19] is explained by tractability of the dynamics of moments of the system observables, which is closely related to the invariance of the class of Gaussian quantum states [15] of the linear system subject to external fields in the vacuum state [14]. This allows the linear quantum control to inherit at least some features of the classical linear control schemes [8], including, in particular, practical computability of quadratic costs. ∗ This work was supported by the Australian Research Council (ARC) and Air Force Office of Scientific Research (AFOSR). This material is based on research sponsored by the Air Force Research Laboratory, under agreement number FA2386-09-1-4089. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the U.S. Government. † School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia: [email protected], [email protected].
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
Ian R. Petersen†
In the present paper, we consider more complicated system-field interactions, described by coupling operators which are quadratic polynomials of the system observables. In combination with a cubic system Hamiltonian, this leads to a novel class of quasilinear quantum stochastic systems, where both the drift and the dispersion matrix of the governing QSDE are affine functions of the system variables. The system-field energetics model imposes certain constraints on the parameters of the QSDE, which are counterparts to the physical realizability conditions of the linear quantum case [6], [13] in the nonlinear setting under consideration. The resulting quasilinear systems extend the linear quantum dynamics and yet retain algebraic closedness in the evolution of mixed moments of the observables up to any order, similarly to their classical analogues [22]. Although such systems no longer preserve Gaussian quantum states, the dynamics of the moments are still amenable to exact analysis, which includes the computation of their steady-state values. In particular, this allows a generalized criterion to be developed for quadratic stochastic stability of the quasilinear systems. Robust stability of a different class of quantum systems with sector-bounded nonlinearities in coupling operators has recently been studied in [17]. The results of the present paper (whose full version with proofs is provided in [20]) can find applications for the generation of non-Gaussian quantum states [23], [24] via synthesis of stable quasilinear quantum systems with manageable moment dynamics. Manipulation of non-Gaussian states is important for quantum information processing with continuous variables which are implementable using quantum optical circuits [3]. II. O PEN QUANTUM SYSTEMS We will briefly review the underlying class of models for open quantum systems which is based on QSDEs. Suppose W (t) := (Wk (t))16k6m is an m-dimensional quantum Wiener process of time t > 0 on a boson Fock space F [14], with the quantum Ito table dW dW T := (dW j dWk )16 j,k6m = Ωdt, (1) where the time argument of W is omitted for brevity, and Ω := (ω jk )16 j,k6m is a constant complex positive semidefinite Hermitian matrix of order m. Here, a complex number ϕ is identified with the operator ϕIF , where IF denotes the identity operator on F . Also, vectors are organised as columns unless indicated otherwise, and the transpose (·)T acts on matrices with operator-valued entries as if the latter were scalars. The real part V := (v jk )16 j,k6m := ReΩ, (2)
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of the matrix Ω is a positive semi-definite symmetric matrix. The entries W1 (t), . . . ,Wm (t) of the vector W (t) are selfadjoint operators on F , associated with the annihilation and creation operator processes of external boson fields. The imaginary part of the quantum Ito matrix Ω in (1) describes the CCRs between the fields in the sense that [dW, dW T ] := ([dW j , dWk ])16 j,k6m = iJdt,
J := 2ImΩ, (3)
where [A, B] := AB − BA is the commutator of operators √ which applies entrywise, and i := −1 is the imaginary unit. We consider an open quantum system (further referred to as the plant) whose state variables X1 (0), . . . , Xn (0) at time t = 0 are self-adjoint operators on a complex separable Hilbert space H . Any such operator ξ can be identified with its ampliation ξ ⊗ IF on the tensor product space H ⊗ F . Suppose the plant interacts only with the external fields, so that the density operator ρ(t), which describes the quantum state [12] of the plant-field system, isolated from the environment, on H ⊗F at time t, evolves in the Schr¨odinger picture of quantum dynamics as ρ(t) = U(t)ρ(0)U(t)† . (4) Here, U(t) is a unitary operator on the Hilbert space H ⊗F , initialized at the identity operator U(0) = IH ⊗F , and (·)† is the operator adjoint. The initial quantum state of the plantfield composite system is assumed to be the tensor product ρ(0) := ϖ(0) ⊗ υ of the initial plant state ϖ(0) on H and the pure state υ := |0ih0| of the external field associated with the vacuum vector |0i in F , where use has been made of the Dirac bra-ket notation [12]. In the Heisenberg picture, an observable ξ (t) on H ⊗ F evolves in a dual unitary fashion to (4): ξ (t) = U(t)† ξ (0)U(t), (5) with the duality being understood in the sense of the equivalence Eξ (t) := Tr(ρ(0)ξ (t)) = Tr(ρ(t)ξ (0)) (6) between two representations of the quantum expectation. The unitary operator U(t) itself is driven by the internal dynamics of the plant (which the plant would have in isolation from the surroundings) and by the plant-field interaction. In the weak interaction limit, which neglects the influence of the plant on the Markov structure of the field in the vacuum state, a wide class of open quantum systems is captured by the following QSDE: dU = −((iH + hT Ωh/2)dt + ihT dW )U. (7) Here, H is the plant Hamiltonian, and h := (h j )16 j6m is a vector of plant-field coupling operators, with h j pertaining to the interaction between the plant and the jth external field. Both H and h1 , . . . , hm are self-adjoint operators on H , which are usually functions of the plant observables X1 (0), . . . , Xn (0). The term hT dW = ∑m k=1 hk dWk in (7), which can be interpreted as an incremental perturbation to the plant Hamiltonian H due to the interaction with the external fields, is an alternative form of the more traditional representation hT dW = i(LT dA # − L† dA ) (8) through the m/2-dimensional field annihilation and creation operator processes A (t) and A (t)# on the Fock space F with the quantum Ito table
A d A # d[A † † ] := dA
A T]
I = m/2 0
0 dt, 0
(9)
† − dA # dA T
so that [dA , dA dA = Im/2 dt. Here, m is assumed to be even, (·)† := ((·)# )T denotes the transpose of the entrywise adjoint (·)# , and Ir is the identity matrix of order r. In application to ordinary matrices, (·)† reduces to the complex conjugate transpose (·)∗ := ((·))T . The representation (8) corresponds to the case when the scattering matrix [14] is the identity matrix. The vector L := (Lk )16k6m/2 consists of linear operators on H , which are not necessarily self-adjoint. The relation of h, W with L, A is described by 1 L i −i h= √ ⊗ Im/2 L# , (10) 1 1 2 1 A 1 1 W=√ ⊗ Im/2 A # , (11) 2 −i i where ⊗ denotes the Kronecker product of matrices, so that the quantum Ito matrix Ω in (1) is corresponding h i 0 1 Ω = Im + i −1 0 ⊗ Im/2 2. In general, the operator hT Ωh := ∑mj,k=1 ω jk h j hk , which takes the form hT Ωh = m/2 L† L = ∑k=1 Lk† Lk in the case (9)–(11), is self-adjoint since T † (h Ωh) = ∑mj,k=1 ω jk hk h j = hT Ω∗ h = hT Ωh. The formulation using h and W , in principle, allows W to be an additive mixture of the quantum noise and a classical random noise such as the standard Wiener process. In view of (3), the “magnitude” of ImΩ = J/2 in comparison with ReΩ (measured, for example, by the spectral radius r(ImΩ(ReΩ)−1 ) which is well defined and is strictly less than one if Ω 0) indicates the relative amount of “quantumness” in W . This setting reduces to the classical noise situation if the matrix Ω in (1) is real, in which case U(t) becomes a random process with values among unitary operators on H ⊗ F ; see [7] and [14, pp. 258–260]. The general situation is treated by applying the quantum Ito rule d(ηζ ) = (dη)ζ + ηdζ + (dη)dζ and using (7) along with the unitarity of U(t) and commutativity between the forward increment dW and the adapted processes. This yields the following QSDE for the density operator ρ(t) in (4): dρ = −i([H, ρ]dt + [hT , ρ]dW ) + tr(ΩTC(ρ))dt, (12) which is referred to as the stochastic quantum master equation [2] in the Schr¨odinger picture. Here, use is made of a self-adjoint operator tr(ΩTC(ρ)) := ∑mj,k=1 ω jkC jk (ρ) on H ⊗ F , with tr(·) denoting a “symbolic” trace (to be distinguished from the complex-valued trace Tr(·) of an operator), where the matrix C(ρ) := (C jk (ρ))16 j,k6m has operator-valued entries C jk (ρ) := hk ρh j − (h j hk ρ + ρh j hk )/2. (13) Note that C jk (ρ)† = Ck j (ρ) for any self-adjoint operator ρ in view of self-adjointness of h1 , . . . , hm , and this, together with Ω∗ = Ω, ensures the self-adjointness of tr(ΩTC(ρ)). III. D ECOHERENCE OPERATOR In the Heisenberg picture, an observable ξ (t) on the composite Hilbert space H ⊗ F , which undergoes the evolution (5), satisfies the QSDE dξ = i([H, ξ ]dt + [h, ξ ]T dW ) + L (ξ )dt. (14)
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Here, both the plant Hamiltonian H(t) and the vector h(t) of plant-field coupling operators are also evolved by the flow (5). However, they depend on the vector X(t) := (Xk (t))16k6n of the plant observables in the same way as H(0) and h(0) do on X(0). Also, L denotes the Gorini-KossakowskiSudarshan-Lindblad (GKSL) superoperator defined by L (ξ ) := tr(ΩT D(ξ )) =
m
∑
ω jk D jk (ξ ),
(15)
j,k=1
where D(ξ ) := (D jk (ξ ))16 j,k6m is a matrix with operatorvalued entries D jk (ξ ) := h j ξ hk − (h j hk ξ + ξ h j hk )/2 = (h j [ξ , hk ] + [h j , ξ ]hk )/2.
(16)
The superoperators D jk are dual to C jk from (13) in the sense that Tr(C jk (ρ)ξ ) = Tr(ρD jk (ξ )). The superoperator matrix D acts on an observable ξ as D(ξ ) = (h[ξ , hT ] + [h, ξ ]hT )/2. The superoperators C jk and their duals D jk , defined by (13), (16), play an important role in the generators of quantum dynamical semigroups [4], [9]. One of such semigroups governs the evolution of the reduced plant density operator ϖ(t) obtained by “tracing out” the quantum noise in (12) over the field vacuum state υ which yields an ODE ϖ˙ = −i[H, ϖ] + tr(ΩTC(ϖ)). The generator of the corresponding semigroup in the dual Heisenberg picture is i[H, ·] + L , where the superoperator L , given by (15), is responsible for decoherence [2] understood as the deviation from a unitary evolution which the plant observables would have alone in the absence of interaction with the environment. It is convenient to apply the QSDE (14) entrywise to the vector X(t) of plant observables as
IV. L INEAR PLANT- FIELD COUPLING Omitting the time dependence, suppose the plant observables X1 , . . . , Xn , which are assembled into the vector X, satisfy CCRs [X, X T ] := ([X j , Xk ])16 j,k6n = iΘ, (21) where Θ := (θ jk )16 j,k6n is a constant real antisymmetric matrix of order n (we denote the space of such matrices by An ). In this case, if the plant-field coupling operators h1 , . . . , hm are polynomials of degree r in the plant observables, then the entries of the dispersion matrix G in (18) and the vector L (X) in (19) are polynomials of degrees r − 1 and 2r − 1, respectively. This property follows from the reduction of a polynomial degree under taking the commutator with the observables due to the CCRs (21): r
[Ξk , X` ] = i ∑ θk j ` Ξk1 ...k j−1 k j+1 ...kr , where
(22)
j=1
(20)
Ξk := Xk1 × . . . × Xkr (23) denotes a degree r monomial of the plant observables specified by an r-index k := (k1 , . . . , kr ) ∈ {1, . . . , n}r , with the order of multiplication being essential in the noncommutative case. The right-hand side of (22) is a polynomial of degree r − 1. In the plant-field interaction model, which is used in linear quantum control [1], [6], [13], [16], the vector h of coupling operators depends linearly on X in the sense that h := MX (24) for some matrix M ∈ Rm×n . In this case, the dispersion matrix G in (18) becomes a real matrix, since G = −i[X, hT ] = −i[X, X T ]M T = ΘM T , where the bilinearity of the commutator is combined with the CCRs (21). In view of Lemma 1, this allows L (X) to inherit from h the linearity with respect to the plant observables. Lemma 2: In the case of CCRs (21) and linear plant-field coupling (24), the vector X of plant observables satisfies a QSDE dX = (i[H, X] + KX)dt + BdW, (25) where the matrices K ∈ Rn×n and B ∈ Rn×m are related to the matrix J from (3) by K := BJM/2, B := ΘM T . (26) Since K and B are constant matrices, the QSDE (25) may acquire nonlinearity with respect to the plant observables only through a nonquadratic part of the plant Hamiltonian H. Indeed, if H is a quadratic polynomial, that is, n 1 n (27) H := ∑ γk + ∑ r jk X j Xk = (γ + RX/2)T X, 2 j=1 k=1 where γ := (γk )16k6n ∈ Rn is a given real vector, and R := (r jk )16 j,k6n is a given real symmetric matrix of order n (we denote the space of such matrices by Sn ), then the commutator identities [12, Eq. (3.50) on p. 38]) and the CCRs (21) imply that i[H, X] = −i([X, X T ]γ + ([X, X T ]RX − (X T R[X, X T ])T )/2)
In the next section, we will employ Lemma 1 in order to review the computation of the drift vector F and the dispersion matrix G for a class of linear open quantum systems.
= Θ(γ + RX). (28) Since the right-hand side of (28) depends affinely on X, then (25) becomes linear with respect to the plant observables: dX = (AX + Θγ)dt + BdW, A := ΘR + K, (29) which corresponds to an open quantum harmonic oscillator [1], commonly employed in linear quantum control.
dX = Fdt + GdW.
(17)
The n-dimensional drift vector F(t) and the dispersion (n × m)-matrix G(t) of this QSDE, defined by F := i[H, X] + L (X),
G := −i[X, hT ],
(18)
are completely specified by the plant Hamiltonian H, the quantum Ito matrix Ω of the field process W from (1), and the vector h of plant-field coupling operators. Lemma 1: The GKSL superoperator (15), applied to the vector X of plant observables, can be computed in terms of the dispersion matrix G from (18) as m 1 GJh + i ∑ ω jk [h j , gk ] . (19) L (X) = 2 j,k=1 Here, the matrix J is defined by (3), and g1 , . . . , gm denote the columns of G: gk = i[hk , X].
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V. A LGEBRAIC CLOSEDNESS IN MOMENT DYNAMICS The linearity of the QSDE (29) ensures algebraic closedness in the evolution of the mixed moments of the plant observables, defined by µk (t) := EΞk (t) (30) in terms of the quantum expectation (6) applied to the monomials (23). The closedness means that, for any positive integer r and any r-index k, the time derivative µ˙ k can be expressed in terms of the mixed moments of order r and lower. This property is a corollary of the following general result. Lemma 3: For any positive integer r and any r-index k := (k1 , . . . , kr ) ∈ {1, . . . , n}r , the mixed moment µk from (30) for the plant observables, governed by the QSDE (17), satisfies r
µk = ∑ E(Ξk1 ...k j−1 Fk j Ξk j+1 ...kr ) j=1 m
+ ∑ ωsu
∑
E(Ξk1 ...k j−1 gk j s Ξk j+1 ...k`−1 gk` u Ξk`+1 ...kr ).(31)
s,u=1 16 j0 , formed by µ0/ and the mixed moments µk for all possible nr multiindices k ∈ {1 . . . , n}r of orders r = 1, 2, 3, . . ., satisfies a system of linear ODEs µ˙ = Ψµ, where Ψ := (ψkν ) is an infinite-dimensional blocklower triangular matrix. The diagonal block of Ψ associated with the moments of order r is a matrix of order nr . Hence, the solution of the system of ODEs (32) can be represented as µ(t) = eΨt µ(0), provided all the moments of the initial plant state ϖ(0) are finite. The matrix exponential eΨt is practically computable due to the block-lower triangular structure of Ψ. Thus, the algebraic closedness (32) allows the system of linear ODEs for the moments (30) to be integrated (numerically or analytically) recursively with respect to r, starting from the mean values of the plant observables for r = 1. In particular, the mean vector and the quantum covariance matrix α := EX, S := cov(X) = E(XX T ) − αα T , (33) with the latter consisting of central moments of second order, satisfy the ODEs α = Aα + Θγ, (34) S = AS + SAT + BΩBT , which allow the steady-state values of the moments to be found from the appropriate algebraic equations if the matrix A, defined in (29), is Hurwitz. Now, a similar reasoning shows that the moment dynamics (31) retains the algebraic
closedness (32) in a more general case where both the drift vector F and the dispersion matrix G of the QSDE (17) are affine functions of X (in the linear case above, G was constant). This corresponds to a wider class of open quantum systems introduced in the next section. VI. Q UASILINEAR OPEN QUANTUM SYSTEMS Retaining the assumption of Section IV that the plant observables satisfy the CCRs (21), we will now consider a wider class of plant-field interactions, in which the coupling operators h1 , . . . , hm are quadratic polynomials of the plant observables: h j = (M j +Y jT /2)X, Y j := R j X. (35) m×n Here, M j denotes the jth row of a matrix M ∈ R , which describes the linear part of the coupling as in (24), and R1 , . . . , Rm ∈ Sn are given matrices which specify the quadratic part. An equivalent vector-matrix form of (35) is h = (M +Y T /2)X, Y := Y1 . . . Ym , (36) where Y is an (n × m)-matrix with columns Y1 , . . . ,Ym whose entries are linear combinations of the plant observables. In the case of quadratic plant-field coupling (35), an argument, similar to the derivation of (28) from (27), allows the kth column (20) of the dispersion matrix G from (18) to be computed as gk = i[(Mk + X T Rk /2)X, X] = Θ(MkT +Yk ), (37) T T where Mk is the kth column of the matrix M , which, in view of (36), implies that G = Θ(M T +Y ) = B + Θ R1 X . . . Rm X , (38) where the matrix B is defined by (26). Therefore, the entries of G are affine functions of the plant observables. From (35) and (37), it follows that the contribution of the operators i[h j , gk ] =i[(M j +Y jT /2)X, ΘRk X] =iΘRk [(M j +X T R j /2)X, X] = ΘRk Θ(M Tj +Y j ) to the right-hand side of (19) is linear with respect to the plant observables: m
i
∑ j,k=1
m
ω jk [h j , gk ] =
∑
ω jk ΘRk Θ(M Tj + R j X).
j,k=1
Thus, in the case of canonically commuting plant observables and quadratic plant-field coupling (35), the dispersion matrix G is an affine function of X, while L (X), given by (19), is a cubic polynomial of X. The latter property suggests finding a Hamiltonian H in the form of a quartic (degree four) polynomial of the plant observables such that the corresponding cubic polynomial i[H, X] counterbalances the quadratic and cubic terms in L (X), thus making the drift vector F in (18) an affine function of X: F = AX + β , (39) n×n n and β ∈ R . Together with G depending where A ∈ R affinely on X as described by (38), the resulting quantum plant, governed by the QSDE
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dX = (AX + β )dt + Θ(M T +Y )dW m
= (AX + β )dt + Θ ∑ (M Tj + R j X)dW j , j=1
(40)
which we will refer to as a quasilinear open quantum system, retains the algebraic closedness (32) in the moment dynamics (31) as discussed in Section V. In the next section, we will show that the problem of finding such a Hamiltonian H is simplified significantly by taking physical realizability conditions into account.
RELATIONS
The commutator of two observables η(t) and ζ (t) on the product space H ⊗ F inherits the evolution (5) with the unitary matrix U(t): [η(t), ζ (t)] = U(t)† [η(0), ζ (0)]U(t). (41) Hence, if η(0) and ζ (0) satisfy a CCR, that is, if [η(0), ζ (0)] = ϕ is the identity operator IH ⊗F up to a complex multiplier ϕ, then the unitarity of U(t) and (41) imply that [η(t), ζ (t)] = ϕU(t)†U(t) = [η(0), ζ (0)] for all t > 0. Therefore, any CCR between observables on the space H ⊗ F is preserved in time. In particular, if the plant observables X1 (0), . . . , Xn (0) are in CCRs with each other, then, the preservation of these CCRs is a necessary condition for physical realizability (PR) of a QSDE of the form (17). Here, in accordance with [6], [13] in the linear case and [10] for nonlinear systems, PR is understood as existence of a plant-field energetics model, specified by the pair (H, h), which generates the particular drift vector F and dispersion matrix G as described by (18). We will now obtain CCR preservation conditions for the quasilinear quantum plant governed by the QSDE (40) which corresponds to (17) with F, G given by (39), (38), respectively. Let M and R denote linear operators which map an n-dimensional vector u and a matrix T of order n to two matrices of order n:
R(T ) := Θ
m
∑
J jk (M Tj uT Rk + R j uMk )Θ,
(42)
J jk R j T Rk Θ.
(43)
j,k=1 m
∑ j,k=1
The significance of these operators is clarified by the following lemma which is instrumental to the proof of Theorem 1. Lemma 4: The matrix GJGT , associated with the dispersion matrix G in (38), is a constant complex matrix (independent of X and the initial quantum state of the plant) if and only if the operators M , R, defined by (42), (43), both vanish on Rn and Sn , respectively. In this case, GJGT = BJBT − iR(Θ)/2, (44) where the matrix B is defined by (26). Since Θ is antisymmetric, the matrix R(Θ) in (44) does not have to vanish under the assumption of Lemma 4 that R = 0 on Sn . A sufficient condition for this assumption to hold can be obtained by using the vectorization of matrices [11]: m
∑
J jk (ΘR j ) ⊗ (ΘRk ) = 0.
E (u) := Θ V (T ) := Θ
m
∑
v jk (M Tj uT Rk + R j uMk )Θ,
(46)
v jk R j T Rk Θ,
(47)
j,k=1 m
∑ j,k=1
VII. P RESERVATION OF CANONICAL COMMUTATION
M (u) := Θ
(43), linear operators E and V which map an n-dimensional vector u and a matrix T of order n to matrices of order n:
(45)
j,k=1
More precisely, the condition (45) is necessary and sufficient for the operator R in (43) to vanish on the space Rn×n which contains Sn . For what follows, we define, similarly to (42),
where v jk are the entries of the real part of the quantum Ito matrix from (2). Theorem 1: The quasilinear QSDE (40) preserves the CCR matrix Θ of the plant observables from (21) if and only if the linear operators M and R, defined by (42), (43), vanish on Rn and Sn , respectively, and AΘ + ΘAT + BJBT = V (Θ). Therefore, Theorem 1 imposes constraints (in terms of the operators M and R) which the quadratic plant-field coupling operators (35) have to satisfy in order to make an affine drift term of the QSDE (40) achievable through an appropriate choice of the plant Hamiltonian H. Lemma 5: Suppose the CCR matrix Θ in (21) is nonsingular, and the operators M and R in (42), (43), associated with the quadratic plant-field coupling model (35), satisfy the conditions of Theorem 1. Then the GKSL vector L (X) in (19) is a quadratic polynomial of the plant observables with the leading term ΘY JMX/4, that is, L (X) = ΘY JMX/4 + (affine function of X). Lemma 5 suggests that the class of candidate plant polynomials H for counterbalancing the nonlinear terms of the GKSL operator L (X) by i[H, X] (to achieve an affine drift vector in the governing QSDE) can be reduced to cubic polynomials. One of such Hamiltonians is found in the next section. VIII. C UBIC PLANT H AMILTONIAN The following theorem provides a characterization of the class of quasilinear quantum stochastic plants described by the QSDE (40) associated with the quadratic plant-field coupling model (35). For its formulation, we introduce a Hamiltonian H := γ T X + X T R0 X/2 − X TY JMX/12, (48) which is a cubic polynomial of the plant observables. Here, γ ∈ Rn and R0 ∈ Sn are arbitrary vector and matrix which part of H, while X TY JMX = T specify theT quadratic X R1 X . . . X Rm X JMX is a homogeneous cubic polynomial of X, specified by the parameters M ∈ Rm×n and R1 , . . . , Rm ∈ Sn of (35)–(36). Theorem 2: Suppose the observables of the open quantum plant under consideration have a nonsingular CCR matrix Θ in (21), and the plant-field coupling is described by the quadratic model (35) whose parameters satisfy the conditions of Theorem 1. Then the cubic plant Hamiltonian H, described by (48), is self-adjoint and leads to a quasilinear QSDE (40).
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Thus, Theorem 2 constructs a physically realizable quasilinear quantum stochastic plant from an appropriately constrained quadratic plant-field coupling model and the corresponding cubic plant Hamiltonian with an arbitrary quadratic part. The matrix A in (40), whose calculation is omitted for the sake of brevity, can be recovered from the proofs of Lemma 5 and Theorem 2. IX. Q UADRATIC STOCHASTIC STABILITY Due to the algebraic closedness in the moment dynamics (32), which extends from the linear case of Section IV to the quasilinear quantum systems (40), the stochastic stability of such systems is amenable to analysis at the level of moments of arbitrarily high order. We will discuss the quadratic stability which is concerned with the first two moments. Theorem 3: For the quasilinear quantum stochastic plant governed by the QSDE (40) and satisfying the CCR preservation conditions of Theorem 1, the mean vector α from (33), and the real part Σ := ReS of the quantum covariance matrix satisfy the ODEs
α = Aα + β ,
(49)
Σ = AΣ + ΣA − V (Σ) + BV B + R(Θ)/4 − E (α) − V (αα T ), (50) where the linear operators E and V are defined by (46), (47). Theorem 3 shows that, unlike the mean-covariance dynamics in the linear case (34), the Hurwitz property of the matrix A is sufficient for the stability of the quasilinear quantum plant (40) only at the level of the first order moments. In view of (50), such a plant is quadratically stable if the real parts of the eigenvalues of the linear operator Σ 7→ AΣ + ΣAT − V (Σ), acting on the space Sn , are all negative. The latter, in view of (47), is equivalent to the Hurwitz property of the following matrix of order n(n + 1)/2: m (ΛT Λ)−1 ΛT In ⊗ A + A ⊗ In + ∑ v jk (ΘR j ) ⊗ (ΘRk ) Λ. T
T
j,k=1 2
Here, Λ ∈ Rn ×n(n+1)/2 denotes the “duplication” matrix [11], [18], which relates the full vectorization vec(Σ) = Λvech(Σ) of a matrix Σ ∈ Sn with its half-vectorization vech(Σ) (that is, the column-wise vectorization of the triangular part of Σ below and including the main diagonal). In this case, the steady-state values limt→+∞ α(t) and limt→+∞ Σ(t) are unique solutions of the corresponding algebraic equations obtained by equating the right-hand sides of (49), (50) to zero. Also note that, for the quasilinear quantum plant, the mean vector α influences the evolution of Σ by entering the right-hand side of (50) in a quadratic fashion, whereas the dynamics of the mean and covariances in the linear case (34) are completely decoupled. X. C ONCLUSION We have introduced a novel class of quasilinear open quantum systems with canonically commuting dynamic variables governed by QSDEs whose drift vector and dispersion matrix are affine functions of the system variables. Despite the nonlinearity of their dynamics and non-Gaussian nature
of quantum states, the evolution of mixed moments of the system variables up to any order is algebraically closed, which makes their stability amenable to exact analysis. We have provided a constructive theory of physical realizability of quasilinear systems using quadratic system-field coupling operators and cubic Hamiltonians. A generalized criterion has been outlined for quadratic stochastic stability of the quasilinear systems. The results of the paper can find applications to the generation of non-Gaussian quantum states with manageable moment dynamics and an optimal design of linear quantum controllers for quasilinear quantum plants. R EFERENCES [1] S.C.Edwards, and V.P.Belavkin, Optimal quantum filtering and quantum feedback control, arXiv:quant-ph/0506018v2, August 1, 2005. [2] C.W.Gardiner, and P.Zoller, Quantum Noise. Springer, Berlin, 2004. [3] S.Ghose, and B.C.Sanders, Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps, J. Mod. Opt., vol. 54, no. 6, 2007, pp. 855–869. [4] V.Gorini, A.Kossakowski, E.C.G.Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys., vol. 17, no. 5, 1976, pp. 821–825. [5] M.R.James, and J.E.Gough, Quantum dissipative systems and feedback control design by interconnection, IEEE Trans. Automat. Contr., vol. 55, no. 8, 2008, pp. 1806–1821. [6] M.R.James, H.I.Nurdin, and I.R.Petersen, H ∞ control of linear quantum stochastic systems, IEEE Trans. Automat. Contr., vol. 53, no. 8, 2008, pp. 1787–1803. [7] A.Kossakowski, On quantum statistical mechanics of non-Hamiltonian systems, Rep. Math. Phys., vol. 3, no. 4, 1972, pp. 247–274. [8] H.Kwakernaak, and R.Sivan, Linear Optimal Control Systems, Wiley, New York, 1972. [9] G.Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys., vol. 48, 1976, pp. 119–130. [10] A.I.Maalouf, and I.R.Petersen, On the physical realizability of a class of nonlinear quantum systems, arXiv:1207.5299v1 [math.OC], 23 July 2012. [11] J.R.Magnus, Linear Structures, Oxford University Press, New York, 1988. [12] E.Merzbacher, Quantum Mechanics, 3rd Ed., Wiley, New York, 1998. [13] H.I.Nurdin, M.R.James, and I.R.Petersen, Coherent quantum LQG control, Automatica, vol. 45, 2009, pp. 1837–1846. [14] K.R.Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkh¨auser, Basel, 1992. [15] K.R.Parthasarathy, What is a Gaussian state? Commun. Stoch. Anal., vol. 4, no. 2, 2010, pp. 143–160. [16] I.R.Petersen, Quantum linear systems theory, Proc. 19th Int. Symp. Math. Theor. Networks Syst., Budapest, Hungary, July 5–9, 2010, pp. 2173–2184. [17] I.R.Petersen, V.A.Ugrinovskii, and M.R.James, Robust stability of quantum systems with a nonlinear coupling operator, to appear. [18] R.E.Skelton, T.Iwasaki, and K.M.Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor & Francis, London, 1998. [19] I.G.Vladimirov, and I.R.Petersen, A quasi-separation principle and Newton-like scheme for coherent quantum LQG control, 18th IFAC World Congress, Milan, Italy, 28 August–2 September, 2011, pp. 4721–4727. [20] I.G.Vladimirov, and I.R.Petersen, Characterization and moment stability analysis of quasilinear quantum stochastic systems with quadratic coupling to external fields, arXiv:1205.3269v1 [quant-ph], 15 May 2012. [21] H.M.Wiseman, and G.J.Milburn, Quantum Measurement and Control, Cambridge University Press, 2009. [22] W.M.Wonham, Optimal stationary control of a linear system with state-dependent noise, SIAM J. Control, vol. 5, no. 3, 1967, pp. 486– 500. [23] M.Yanagisawa, Non-Gaussian state generation from linear elements via feedback, Phys. Rev. Lett., vol. 103, no. 20, 2009, pp. 203601-1-4. [24] G.Zhang, and M.R.James, On the response of linear quantum stochastic systems to single-photon inputs and pulse shaping of photon wave packets, Proc. Australian Control Conference, Melbourne, 10–11 November, 2011, pp. 62–67.
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Characterization and Moment Stability Analysis of Quasilinear Quantum Stochastic Systems with Quadratic Coupling to External Fields∗ Igor G. Vladimirov† ,
Abstract— The paper is concerned with open quantum systems whose Heisenberg dynamics are described by quantum stochastic differential equations driven by external boson fields. The system-field coupling operators are assumed to be quadratic polynomials of the system variables, with the latter satisfying canonical commutation relations. In combination with a cubic system Hamiltonian, this leads to quasilinear quantum stochastic systems which extend the class of linear quantum systems and yet retain algebraic closedness in the evolution of mixed moments of system variables up to any order. Although such a system is nonlinear and its quantum state is no longer Gaussian, the dynamics of the moments are amenable to exact analysis, including the computation of their steady-state values. A generalized criterion is outlined for quadratic stability of the quasilinear systems.
I. I NTRODUCTION Open quantum systems, driven by external boson fields, are usually described in the Heisenberg picture by quantum stochastic differential equations (QSDEs) [14]. This class of dynamical systems, whose variables are noncommutative operators on a Hilbert space, is a common source of models in quantum optics [2] and quantum control [1], [5], [6], [21]. Central to this approach is a Markovian description of interaction between the quantum-mechanical system and its environment, whose “energetics” is specified by the system Hamiltonian and system-field coupling operators. The case where the Hamiltonian is quadratic and the coupling operators are linear with respect to the system variables, and the latter satisfy canonical commutation relations (CCRs) [12], corresponds to an open quantum harmonic oscillator. Its importance for linear quantum stochastic control [1], [13], [16], [19] is explained by tractability of the dynamics of moments of the system observables, which is closely related to the invariance of the class of Gaussian quantum states [15] of the linear system subject to external fields in the vacuum state [14]. This allows the linear quantum control to inherit at least some features of the classical linear control schemes [8], including, in particular, practical computability of quadratic costs. ∗ This work was supported by the Australian Research Council (ARC) and Air Force Office of Scientific Research (AFOSR). This material is based on research sponsored by the Air Force Research Laboratory, under agreement number FA2386-09-1-4089. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the U.S. Government. † School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia: [email protected], [email protected].
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
Ian R. Petersen†
In the present paper, we consider more complicated system-field interactions, described by coupling operators which are quadratic polynomials of the system observables. In combination with a cubic system Hamiltonian, this leads to a novel class of quasilinear quantum stochastic systems, where both the drift and the dispersion matrix of the governing QSDE are affine functions of the system variables. The system-field energetics model imposes certain constraints on the parameters of the QSDE, which are counterparts to the physical realizability conditions of the linear quantum case [6], [13] in the nonlinear setting under consideration. The resulting quasilinear systems extend the linear quantum dynamics and yet retain algebraic closedness in the evolution of mixed moments of the observables up to any order, similarly to their classical analogues [22]. Although such systems no longer preserve Gaussian quantum states, the dynamics of the moments are still amenable to exact analysis, which includes the computation of their steady-state values. In particular, this allows a generalized criterion to be developed for quadratic stochastic stability of the quasilinear systems. Robust stability of a different class of quantum systems with sector-bounded nonlinearities in coupling operators has recently been studied in [17]. The results of the present paper (whose full version with proofs is provided in [20]) can find applications for the generation of non-Gaussian quantum states [23], [24] via synthesis of stable quasilinear quantum systems with manageable moment dynamics. Manipulation of non-Gaussian states is important for quantum information processing with continuous variables which are implementable using quantum optical circuits [3]. II. O PEN QUANTUM SYSTEMS We will briefly review the underlying class of models for open quantum systems which is based on QSDEs. Suppose W (t) := (Wk (t))16k6m is an m-dimensional quantum Wiener process of time t > 0 on a boson Fock space F [14], with the quantum Ito table dW dW T := (dW j dWk )16 j,k6m = Ωdt, (1) where the time argument of W is omitted for brevity, and Ω := (ω jk )16 j,k6m is a constant complex positive semidefinite Hermitian matrix of order m. Here, a complex number ϕ is identified with the operator ϕIF , where IF denotes the identity operator on F . Also, vectors are organised as columns unless indicated otherwise, and the transpose (·)T acts on matrices with operator-valued entries as if the latter were scalars. The real part V := (v jk )16 j,k6m := ReΩ, (2)
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of the matrix Ω is a positive semi-definite symmetric matrix. The entries W1 (t), . . . ,Wm (t) of the vector W (t) are selfadjoint operators on F , associated with the annihilation and creation operator processes of external boson fields. The imaginary part of the quantum Ito matrix Ω in (1) describes the CCRs between the fields in the sense that [dW, dW T ] := ([dW j , dWk ])16 j,k6m = iJdt,
J := 2ImΩ, (3)
where [A, B] := AB − BA is the commutator of operators √ which applies entrywise, and i := −1 is the imaginary unit. We consider an open quantum system (further referred to as the plant) whose state variables X1 (0), . . . , Xn (0) at time t = 0 are self-adjoint operators on a complex separable Hilbert space H . Any such operator ξ can be identified with its ampliation ξ ⊗ IF on the tensor product space H ⊗ F . Suppose the plant interacts only with the external fields, so that the density operator ρ(t), which describes the quantum state [12] of the plant-field system, isolated from the environment, on H ⊗F at time t, evolves in the Schr¨odinger picture of quantum dynamics as ρ(t) = U(t)ρ(0)U(t)† . (4) Here, U(t) is a unitary operator on the Hilbert space H ⊗F , initialized at the identity operator U(0) = IH ⊗F , and (·)† is the operator adjoint. The initial quantum state of the plantfield composite system is assumed to be the tensor product ρ(0) := ϖ(0) ⊗ υ of the initial plant state ϖ(0) on H and the pure state υ := |0ih0| of the external field associated with the vacuum vector |0i in F , where use has been made of the Dirac bra-ket notation [12]. In the Heisenberg picture, an observable ξ (t) on H ⊗ F evolves in a dual unitary fashion to (4): ξ (t) = U(t)† ξ (0)U(t), (5) with the duality being understood in the sense of the equivalence Eξ (t) := Tr(ρ(0)ξ (t)) = Tr(ρ(t)ξ (0)) (6) between two representations of the quantum expectation. The unitary operator U(t) itself is driven by the internal dynamics of the plant (which the plant would have in isolation from the surroundings) and by the plant-field interaction. In the weak interaction limit, which neglects the influence of the plant on the Markov structure of the field in the vacuum state, a wide class of open quantum systems is captured by the following QSDE: dU = −((iH + hT Ωh/2)dt + ihT dW )U. (7) Here, H is the plant Hamiltonian, and h := (h j )16 j6m is a vector of plant-field coupling operators, with h j pertaining to the interaction between the plant and the jth external field. Both H and h1 , . . . , hm are self-adjoint operators on H , which are usually functions of the plant observables X1 (0), . . . , Xn (0). The term hT dW = ∑m k=1 hk dWk in (7), which can be interpreted as an incremental perturbation to the plant Hamiltonian H due to the interaction with the external fields, is an alternative form of the more traditional representation hT dW = i(LT dA # − L† dA ) (8) through the m/2-dimensional field annihilation and creation operator processes A (t) and A (t)# on the Fock space F with the quantum Ito table
A d A # d[A † † ] := dA
A T]
I = m/2 0
0 dt, 0
(9)
† − dA # dA T
so that [dA , dA dA = Im/2 dt. Here, m is assumed to be even, (·)† := ((·)# )T denotes the transpose of the entrywise adjoint (·)# , and Ir is the identity matrix of order r. In application to ordinary matrices, (·)† reduces to the complex conjugate transpose (·)∗ := ((·))T . The representation (8) corresponds to the case when the scattering matrix [14] is the identity matrix. The vector L := (Lk )16k6m/2 consists of linear operators on H , which are not necessarily self-adjoint. The relation of h, W with L, A is described by 1 L i −i h= √ ⊗ Im/2 L# , (10) 1 1 2 1 A 1 1 W=√ ⊗ Im/2 A # , (11) 2 −i i where ⊗ denotes the Kronecker product of matrices, so that the quantum Ito matrix Ω in (1) is corresponding h i 0 1 Ω = Im + i −1 0 ⊗ Im/2 2. In general, the operator hT Ωh := ∑mj,k=1 ω jk h j hk , which takes the form hT Ωh = m/2 L† L = ∑k=1 Lk† Lk in the case (9)–(11), is self-adjoint since T † (h Ωh) = ∑mj,k=1 ω jk hk h j = hT Ω∗ h = hT Ωh. The formulation using h and W , in principle, allows W to be an additive mixture of the quantum noise and a classical random noise such as the standard Wiener process. In view of (3), the “magnitude” of ImΩ = J/2 in comparison with ReΩ (measured, for example, by the spectral radius r(ImΩ(ReΩ)−1 ) which is well defined and is strictly less than one if Ω 0) indicates the relative amount of “quantumness” in W . This setting reduces to the classical noise situation if the matrix Ω in (1) is real, in which case U(t) becomes a random process with values among unitary operators on H ⊗ F ; see [7] and [14, pp. 258–260]. The general situation is treated by applying the quantum Ito rule d(ηζ ) = (dη)ζ + ηdζ + (dη)dζ and using (7) along with the unitarity of U(t) and commutativity between the forward increment dW and the adapted processes. This yields the following QSDE for the density operator ρ(t) in (4): dρ = −i([H, ρ]dt + [hT , ρ]dW ) + tr(ΩTC(ρ))dt, (12) which is referred to as the stochastic quantum master equation [2] in the Schr¨odinger picture. Here, use is made of a self-adjoint operator tr(ΩTC(ρ)) := ∑mj,k=1 ω jkC jk (ρ) on H ⊗ F , with tr(·) denoting a “symbolic” trace (to be distinguished from the complex-valued trace Tr(·) of an operator), where the matrix C(ρ) := (C jk (ρ))16 j,k6m has operator-valued entries C jk (ρ) := hk ρh j − (h j hk ρ + ρh j hk )/2. (13) Note that C jk (ρ)† = Ck j (ρ) for any self-adjoint operator ρ in view of self-adjointness of h1 , . . . , hm , and this, together with Ω∗ = Ω, ensures the self-adjointness of tr(ΩTC(ρ)). III. D ECOHERENCE OPERATOR In the Heisenberg picture, an observable ξ (t) on the composite Hilbert space H ⊗ F , which undergoes the evolution (5), satisfies the QSDE dξ = i([H, ξ ]dt + [h, ξ ]T dW ) + L (ξ )dt. (14)
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Here, both the plant Hamiltonian H(t) and the vector h(t) of plant-field coupling operators are also evolved by the flow (5). However, they depend on the vector X(t) := (Xk (t))16k6n of the plant observables in the same way as H(0) and h(0) do on X(0). Also, L denotes the Gorini-KossakowskiSudarshan-Lindblad (GKSL) superoperator defined by L (ξ ) := tr(ΩT D(ξ )) =
m
∑
ω jk D jk (ξ ),
(15)
j,k=1
where D(ξ ) := (D jk (ξ ))16 j,k6m is a matrix with operatorvalued entries D jk (ξ ) := h j ξ hk − (h j hk ξ + ξ h j hk )/2 = (h j [ξ , hk ] + [h j , ξ ]hk )/2.
(16)
The superoperators D jk are dual to C jk from (13) in the sense that Tr(C jk (ρ)ξ ) = Tr(ρD jk (ξ )). The superoperator matrix D acts on an observable ξ as D(ξ ) = (h[ξ , hT ] + [h, ξ ]hT )/2. The superoperators C jk and their duals D jk , defined by (13), (16), play an important role in the generators of quantum dynamical semigroups [4], [9]. One of such semigroups governs the evolution of the reduced plant density operator ϖ(t) obtained by “tracing out” the quantum noise in (12) over the field vacuum state υ which yields an ODE ϖ˙ = −i[H, ϖ] + tr(ΩTC(ϖ)). The generator of the corresponding semigroup in the dual Heisenberg picture is i[H, ·] + L , where the superoperator L , given by (15), is responsible for decoherence [2] understood as the deviation from a unitary evolution which the plant observables would have alone in the absence of interaction with the environment. It is convenient to apply the QSDE (14) entrywise to the vector X(t) of plant observables as
IV. L INEAR PLANT- FIELD COUPLING Omitting the time dependence, suppose the plant observables X1 , . . . , Xn , which are assembled into the vector X, satisfy CCRs [X, X T ] := ([X j , Xk ])16 j,k6n = iΘ, (21) where Θ := (θ jk )16 j,k6n is a constant real antisymmetric matrix of order n (we denote the space of such matrices by An ). In this case, if the plant-field coupling operators h1 , . . . , hm are polynomials of degree r in the plant observables, then the entries of the dispersion matrix G in (18) and the vector L (X) in (19) are polynomials of degrees r − 1 and 2r − 1, respectively. This property follows from the reduction of a polynomial degree under taking the commutator with the observables due to the CCRs (21): r
[Ξk , X` ] = i ∑ θk j ` Ξk1 ...k j−1 k j+1 ...kr , where
(22)
j=1
(20)
Ξk := Xk1 × . . . × Xkr (23) denotes a degree r monomial of the plant observables specified by an r-index k := (k1 , . . . , kr ) ∈ {1, . . . , n}r , with the order of multiplication being essential in the noncommutative case. The right-hand side of (22) is a polynomial of degree r − 1. In the plant-field interaction model, which is used in linear quantum control [1], [6], [13], [16], the vector h of coupling operators depends linearly on X in the sense that h := MX (24) for some matrix M ∈ Rm×n . In this case, the dispersion matrix G in (18) becomes a real matrix, since G = −i[X, hT ] = −i[X, X T ]M T = ΘM T , where the bilinearity of the commutator is combined with the CCRs (21). In view of Lemma 1, this allows L (X) to inherit from h the linearity with respect to the plant observables. Lemma 2: In the case of CCRs (21) and linear plant-field coupling (24), the vector X of plant observables satisfies a QSDE dX = (i[H, X] + KX)dt + BdW, (25) where the matrices K ∈ Rn×n and B ∈ Rn×m are related to the matrix J from (3) by K := BJM/2, B := ΘM T . (26) Since K and B are constant matrices, the QSDE (25) may acquire nonlinearity with respect to the plant observables only through a nonquadratic part of the plant Hamiltonian H. Indeed, if H is a quadratic polynomial, that is, n 1 n (27) H := ∑ γk + ∑ r jk X j Xk = (γ + RX/2)T X, 2 j=1 k=1 where γ := (γk )16k6n ∈ Rn is a given real vector, and R := (r jk )16 j,k6n is a given real symmetric matrix of order n (we denote the space of such matrices by Sn ), then the commutator identities [12, Eq. (3.50) on p. 38]) and the CCRs (21) imply that i[H, X] = −i([X, X T ]γ + ([X, X T ]RX − (X T R[X, X T ])T )/2)
In the next section, we will employ Lemma 1 in order to review the computation of the drift vector F and the dispersion matrix G for a class of linear open quantum systems.
= Θ(γ + RX). (28) Since the right-hand side of (28) depends affinely on X, then (25) becomes linear with respect to the plant observables: dX = (AX + Θγ)dt + BdW, A := ΘR + K, (29) which corresponds to an open quantum harmonic oscillator [1], commonly employed in linear quantum control.
dX = Fdt + GdW.
(17)
The n-dimensional drift vector F(t) and the dispersion (n × m)-matrix G(t) of this QSDE, defined by F := i[H, X] + L (X),
G := −i[X, hT ],
(18)
are completely specified by the plant Hamiltonian H, the quantum Ito matrix Ω of the field process W from (1), and the vector h of plant-field coupling operators. Lemma 1: The GKSL superoperator (15), applied to the vector X of plant observables, can be computed in terms of the dispersion matrix G from (18) as m 1 GJh + i ∑ ω jk [h j , gk ] . (19) L (X) = 2 j,k=1 Here, the matrix J is defined by (3), and g1 , . . . , gm denote the columns of G: gk = i[hk , X].
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V. A LGEBRAIC CLOSEDNESS IN MOMENT DYNAMICS The linearity of the QSDE (29) ensures algebraic closedness in the evolution of the mixed moments of the plant observables, defined by µk (t) := EΞk (t) (30) in terms of the quantum expectation (6) applied to the monomials (23). The closedness means that, for any positive integer r and any r-index k, the time derivative µ˙ k can be expressed in terms of the mixed moments of order r and lower. This property is a corollary of the following general result. Lemma 3: For any positive integer r and any r-index k := (k1 , . . . , kr ) ∈ {1, . . . , n}r , the mixed moment µk from (30) for the plant observables, governed by the QSDE (17), satisfies r
µk = ∑ E(Ξk1 ...k j−1 Fk j Ξk j+1 ...kr ) j=1 m
+ ∑ ωsu
∑
E(Ξk1 ...k j−1 gk j s Ξk j+1 ...k`−1 gk` u Ξk`+1 ...kr ).(31)
s,u=1 16 j0 , formed by µ0/ and the mixed moments µk for all possible nr multiindices k ∈ {1 . . . , n}r of orders r = 1, 2, 3, . . ., satisfies a system of linear ODEs µ˙ = Ψµ, where Ψ := (ψkν ) is an infinite-dimensional blocklower triangular matrix. The diagonal block of Ψ associated with the moments of order r is a matrix of order nr . Hence, the solution of the system of ODEs (32) can be represented as µ(t) = eΨt µ(0), provided all the moments of the initial plant state ϖ(0) are finite. The matrix exponential eΨt is practically computable due to the block-lower triangular structure of Ψ. Thus, the algebraic closedness (32) allows the system of linear ODEs for the moments (30) to be integrated (numerically or analytically) recursively with respect to r, starting from the mean values of the plant observables for r = 1. In particular, the mean vector and the quantum covariance matrix α := EX, S := cov(X) = E(XX T ) − αα T , (33) with the latter consisting of central moments of second order, satisfy the ODEs α = Aα + Θγ, (34) S = AS + SAT + BΩBT , which allow the steady-state values of the moments to be found from the appropriate algebraic equations if the matrix A, defined in (29), is Hurwitz. Now, a similar reasoning shows that the moment dynamics (31) retains the algebraic
closedness (32) in a more general case where both the drift vector F and the dispersion matrix G of the QSDE (17) are affine functions of X (in the linear case above, G was constant). This corresponds to a wider class of open quantum systems introduced in the next section. VI. Q UASILINEAR OPEN QUANTUM SYSTEMS Retaining the assumption of Section IV that the plant observables satisfy the CCRs (21), we will now consider a wider class of plant-field interactions, in which the coupling operators h1 , . . . , hm are quadratic polynomials of the plant observables: h j = (M j +Y jT /2)X, Y j := R j X. (35) m×n Here, M j denotes the jth row of a matrix M ∈ R , which describes the linear part of the coupling as in (24), and R1 , . . . , Rm ∈ Sn are given matrices which specify the quadratic part. An equivalent vector-matrix form of (35) is h = (M +Y T /2)X, Y := Y1 . . . Ym , (36) where Y is an (n × m)-matrix with columns Y1 , . . . ,Ym whose entries are linear combinations of the plant observables. In the case of quadratic plant-field coupling (35), an argument, similar to the derivation of (28) from (27), allows the kth column (20) of the dispersion matrix G from (18) to be computed as gk = i[(Mk + X T Rk /2)X, X] = Θ(MkT +Yk ), (37) T T where Mk is the kth column of the matrix M , which, in view of (36), implies that G = Θ(M T +Y ) = B + Θ R1 X . . . Rm X , (38) where the matrix B is defined by (26). Therefore, the entries of G are affine functions of the plant observables. From (35) and (37), it follows that the contribution of the operators i[h j , gk ] =i[(M j +Y jT /2)X, ΘRk X] =iΘRk [(M j +X T R j /2)X, X] = ΘRk Θ(M Tj +Y j ) to the right-hand side of (19) is linear with respect to the plant observables: m
i
∑ j,k=1
m
ω jk [h j , gk ] =
∑
ω jk ΘRk Θ(M Tj + R j X).
j,k=1
Thus, in the case of canonically commuting plant observables and quadratic plant-field coupling (35), the dispersion matrix G is an affine function of X, while L (X), given by (19), is a cubic polynomial of X. The latter property suggests finding a Hamiltonian H in the form of a quartic (degree four) polynomial of the plant observables such that the corresponding cubic polynomial i[H, X] counterbalances the quadratic and cubic terms in L (X), thus making the drift vector F in (18) an affine function of X: F = AX + β , (39) n×n n and β ∈ R . Together with G depending where A ∈ R affinely on X as described by (38), the resulting quantum plant, governed by the QSDE
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dX = (AX + β )dt + Θ(M T +Y )dW m
= (AX + β )dt + Θ ∑ (M Tj + R j X)dW j , j=1
(40)
which we will refer to as a quasilinear open quantum system, retains the algebraic closedness (32) in the moment dynamics (31) as discussed in Section V. In the next section, we will show that the problem of finding such a Hamiltonian H is simplified significantly by taking physical realizability conditions into account.
RELATIONS
The commutator of two observables η(t) and ζ (t) on the product space H ⊗ F inherits the evolution (5) with the unitary matrix U(t): [η(t), ζ (t)] = U(t)† [η(0), ζ (0)]U(t). (41) Hence, if η(0) and ζ (0) satisfy a CCR, that is, if [η(0), ζ (0)] = ϕ is the identity operator IH ⊗F up to a complex multiplier ϕ, then the unitarity of U(t) and (41) imply that [η(t), ζ (t)] = ϕU(t)†U(t) = [η(0), ζ (0)] for all t > 0. Therefore, any CCR between observables on the space H ⊗ F is preserved in time. In particular, if the plant observables X1 (0), . . . , Xn (0) are in CCRs with each other, then, the preservation of these CCRs is a necessary condition for physical realizability (PR) of a QSDE of the form (17). Here, in accordance with [6], [13] in the linear case and [10] for nonlinear systems, PR is understood as existence of a plant-field energetics model, specified by the pair (H, h), which generates the particular drift vector F and dispersion matrix G as described by (18). We will now obtain CCR preservation conditions for the quasilinear quantum plant governed by the QSDE (40) which corresponds to (17) with F, G given by (39), (38), respectively. Let M and R denote linear operators which map an n-dimensional vector u and a matrix T of order n to two matrices of order n:
R(T ) := Θ
m
∑
J jk (M Tj uT Rk + R j uMk )Θ,
(42)
J jk R j T Rk Θ.
(43)
j,k=1 m
∑ j,k=1
The significance of these operators is clarified by the following lemma which is instrumental to the proof of Theorem 1. Lemma 4: The matrix GJGT , associated with the dispersion matrix G in (38), is a constant complex matrix (independent of X and the initial quantum state of the plant) if and only if the operators M , R, defined by (42), (43), both vanish on Rn and Sn , respectively. In this case, GJGT = BJBT − iR(Θ)/2, (44) where the matrix B is defined by (26). Since Θ is antisymmetric, the matrix R(Θ) in (44) does not have to vanish under the assumption of Lemma 4 that R = 0 on Sn . A sufficient condition for this assumption to hold can be obtained by using the vectorization of matrices [11]: m
∑
J jk (ΘR j ) ⊗ (ΘRk ) = 0.
E (u) := Θ V (T ) := Θ
m
∑
v jk (M Tj uT Rk + R j uMk )Θ,
(46)
v jk R j T Rk Θ,
(47)
j,k=1 m
∑ j,k=1
VII. P RESERVATION OF CANONICAL COMMUTATION
M (u) := Θ
(43), linear operators E and V which map an n-dimensional vector u and a matrix T of order n to matrices of order n:
(45)
j,k=1
More precisely, the condition (45) is necessary and sufficient for the operator R in (43) to vanish on the space Rn×n which contains Sn . For what follows, we define, similarly to (42),
where v jk are the entries of the real part of the quantum Ito matrix from (2). Theorem 1: The quasilinear QSDE (40) preserves the CCR matrix Θ of the plant observables from (21) if and only if the linear operators M and R, defined by (42), (43), vanish on Rn and Sn , respectively, and AΘ + ΘAT + BJBT = V (Θ). Therefore, Theorem 1 imposes constraints (in terms of the operators M and R) which the quadratic plant-field coupling operators (35) have to satisfy in order to make an affine drift term of the QSDE (40) achievable through an appropriate choice of the plant Hamiltonian H. Lemma 5: Suppose the CCR matrix Θ in (21) is nonsingular, and the operators M and R in (42), (43), associated with the quadratic plant-field coupling model (35), satisfy the conditions of Theorem 1. Then the GKSL vector L (X) in (19) is a quadratic polynomial of the plant observables with the leading term ΘY JMX/4, that is, L (X) = ΘY JMX/4 + (affine function of X). Lemma 5 suggests that the class of candidate plant polynomials H for counterbalancing the nonlinear terms of the GKSL operator L (X) by i[H, X] (to achieve an affine drift vector in the governing QSDE) can be reduced to cubic polynomials. One of such Hamiltonians is found in the next section. VIII. C UBIC PLANT H AMILTONIAN The following theorem provides a characterization of the class of quasilinear quantum stochastic plants described by the QSDE (40) associated with the quadratic plant-field coupling model (35). For its formulation, we introduce a Hamiltonian H := γ T X + X T R0 X/2 − X TY JMX/12, (48) which is a cubic polynomial of the plant observables. Here, γ ∈ Rn and R0 ∈ Sn are arbitrary vector and matrix which part of H, while X TY JMX = T specify theT quadratic X R1 X . . . X Rm X JMX is a homogeneous cubic polynomial of X, specified by the parameters M ∈ Rm×n and R1 , . . . , Rm ∈ Sn of (35)–(36). Theorem 2: Suppose the observables of the open quantum plant under consideration have a nonsingular CCR matrix Θ in (21), and the plant-field coupling is described by the quadratic model (35) whose parameters satisfy the conditions of Theorem 1. Then the cubic plant Hamiltonian H, described by (48), is self-adjoint and leads to a quasilinear QSDE (40).
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Thus, Theorem 2 constructs a physically realizable quasilinear quantum stochastic plant from an appropriately constrained quadratic plant-field coupling model and the corresponding cubic plant Hamiltonian with an arbitrary quadratic part. The matrix A in (40), whose calculation is omitted for the sake of brevity, can be recovered from the proofs of Lemma 5 and Theorem 2. IX. Q UADRATIC STOCHASTIC STABILITY Due to the algebraic closedness in the moment dynamics (32), which extends from the linear case of Section IV to the quasilinear quantum systems (40), the stochastic stability of such systems is amenable to analysis at the level of moments of arbitrarily high order. We will discuss the quadratic stability which is concerned with the first two moments. Theorem 3: For the quasilinear quantum stochastic plant governed by the QSDE (40) and satisfying the CCR preservation conditions of Theorem 1, the mean vector α from (33), and the real part Σ := ReS of the quantum covariance matrix satisfy the ODEs
α = Aα + β ,
(49)
Σ = AΣ + ΣA − V (Σ) + BV B + R(Θ)/4 − E (α) − V (αα T ), (50) where the linear operators E and V are defined by (46), (47). Theorem 3 shows that, unlike the mean-covariance dynamics in the linear case (34), the Hurwitz property of the matrix A is sufficient for the stability of the quasilinear quantum plant (40) only at the level of the first order moments. In view of (50), such a plant is quadratically stable if the real parts of the eigenvalues of the linear operator Σ 7→ AΣ + ΣAT − V (Σ), acting on the space Sn , are all negative. The latter, in view of (47), is equivalent to the Hurwitz property of the following matrix of order n(n + 1)/2: m (ΛT Λ)−1 ΛT In ⊗ A + A ⊗ In + ∑ v jk (ΘR j ) ⊗ (ΘRk ) Λ. T
T
j,k=1 2
Here, Λ ∈ Rn ×n(n+1)/2 denotes the “duplication” matrix [11], [18], which relates the full vectorization vec(Σ) = Λvech(Σ) of a matrix Σ ∈ Sn with its half-vectorization vech(Σ) (that is, the column-wise vectorization of the triangular part of Σ below and including the main diagonal). In this case, the steady-state values limt→+∞ α(t) and limt→+∞ Σ(t) are unique solutions of the corresponding algebraic equations obtained by equating the right-hand sides of (49), (50) to zero. Also note that, for the quasilinear quantum plant, the mean vector α influences the evolution of Σ by entering the right-hand side of (50) in a quadratic fashion, whereas the dynamics of the mean and covariances in the linear case (34) are completely decoupled. X. C ONCLUSION We have introduced a novel class of quasilinear open quantum systems with canonically commuting dynamic variables governed by QSDEs whose drift vector and dispersion matrix are affine functions of the system variables. Despite the nonlinearity of their dynamics and non-Gaussian nature
of quantum states, the evolution of mixed moments of the system variables up to any order is algebraically closed, which makes their stability amenable to exact analysis. We have provided a constructive theory of physical realizability of quasilinear systems using quadratic system-field coupling operators and cubic Hamiltonians. A generalized criterion has been outlined for quadratic stochastic stability of the quasilinear systems. The results of the paper can find applications to the generation of non-Gaussian quantum states with manageable moment dynamics and an optimal design of linear quantum controllers for quasilinear quantum plants. R EFERENCES [1] S.C.Edwards, and V.P.Belavkin, Optimal quantum filtering and quantum feedback control, arXiv:quant-ph/0506018v2, August 1, 2005. [2] C.W.Gardiner, and P.Zoller, Quantum Noise. Springer, Berlin, 2004. [3] S.Ghose, and B.C.Sanders, Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps, J. Mod. Opt., vol. 54, no. 6, 2007, pp. 855–869. [4] V.Gorini, A.Kossakowski, E.C.G.Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys., vol. 17, no. 5, 1976, pp. 821–825. [5] M.R.James, and J.E.Gough, Quantum dissipative systems and feedback control design by interconnection, IEEE Trans. Automat. Contr., vol. 55, no. 8, 2008, pp. 1806–1821. [6] M.R.James, H.I.Nurdin, and I.R.Petersen, H ∞ control of linear quantum stochastic systems, IEEE Trans. Automat. Contr., vol. 53, no. 8, 2008, pp. 1787–1803. [7] A.Kossakowski, On quantum statistical mechanics of non-Hamiltonian systems, Rep. Math. Phys., vol. 3, no. 4, 1972, pp. 247–274. [8] H.Kwakernaak, and R.Sivan, Linear Optimal Control Systems, Wiley, New York, 1972. [9] G.Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys., vol. 48, 1976, pp. 119–130. [10] A.I.Maalouf, and I.R.Petersen, On the physical realizability of a class of nonlinear quantum systems, arXiv:1207.5299v1 [math.OC], 23 July 2012. [11] J.R.Magnus, Linear Structures, Oxford University Press, New York, 1988. [12] E.Merzbacher, Quantum Mechanics, 3rd Ed., Wiley, New York, 1998. [13] H.I.Nurdin, M.R.James, and I.R.Petersen, Coherent quantum LQG control, Automatica, vol. 45, 2009, pp. 1837–1846. [14] K.R.Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkh¨auser, Basel, 1992. [15] K.R.Parthasarathy, What is a Gaussian state? Commun. Stoch. Anal., vol. 4, no. 2, 2010, pp. 143–160. [16] I.R.Petersen, Quantum linear systems theory, Proc. 19th Int. Symp. Math. Theor. Networks Syst., Budapest, Hungary, July 5–9, 2010, pp. 2173–2184. [17] I.R.Petersen, V.A.Ugrinovskii, and M.R.James, Robust stability of quantum systems with a nonlinear coupling operator, to appear. [18] R.E.Skelton, T.Iwasaki, and K.M.Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor & Francis, London, 1998. [19] I.G.Vladimirov, and I.R.Petersen, A quasi-separation principle and Newton-like scheme for coherent quantum LQG control, 18th IFAC World Congress, Milan, Italy, 28 August–2 September, 2011, pp. 4721–4727. [20] I.G.Vladimirov, and I.R.Petersen, Characterization and moment stability analysis of quasilinear quantum stochastic systems with quadratic coupling to external fields, arXiv:1205.3269v1 [quant-ph], 15 May 2012. [21] H.M.Wiseman, and G.J.Milburn, Quantum Measurement and Control, Cambridge University Press, 2009. [22] W.M.Wonham, Optimal stationary control of a linear system with state-dependent noise, SIAM J. Control, vol. 5, no. 3, 1967, pp. 486– 500. [23] M.Yanagisawa, Non-Gaussian state generation from linear elements via feedback, Phys. Rev. Lett., vol. 103, no. 20, 2009, pp. 203601-1-4. [24] G.Zhang, and M.R.James, On the response of linear quantum stochastic systems to single-photon inputs and pulse shaping of photon wave packets, Proc. Australian Control Conference, Melbourne, 10–11 November, 2011, pp. 62–67.
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