On the generalization of linear least mean squares estimation to quantum systems with non-commutative outputs

The purpose of this paper is to study the problem of generalizing the Belavkin-Kalman filter to the case where the classical measurement signal is replaced by a fully quantum non-commutative output signal. We formulate a least mean squares estimation problem that involves a non-commutative system as the filter processing the non-commutative output signal. We solve this estimation problem within the framework of non-commutative probability. Also, we find the necessary and sufficient conditions which make these non-commutative estimators physically realizable. These conditions are restrictive in practice.


Introduction
Quantum filtering theory as a fundamental theory in quantum optics, with its origins in the work of Davis in the 1960s [8,9] concerning open quantum systems and generalized measurement theory, and culminating in the general theory developed by Belavkin during the 1980's [1,4,3]. The quantum filter is a stochastic differential equation for the conditional state, from which the best estimates of the system observables may be obtained. In related work by Carmichael, the quantum filter is referred to as the stochastic master equation [7,30].
One application of the quantum filter, or variants of it, is in measurement feedback control [5,10,13,30,15]. As in classical control theory, optimal measurement feedback control strategies may be expressed as functions of information states, of which the conditional state is a particular case [13,14]. However, feedback control of quantum systems need not involve measurements, and indeed the topic of coherent quantum feedback is evolving [22,19,29,12,31,18,14], though a general theory of optimal design of coherent quantum feedback systems is at present unavailable. In coherent quantum feedback control, the controller is also a quantum system, and information flowing in the feedback loop is also quantum (e.g. via a quantum field). This type of feedback has many applications for quantum memories and quantum error correction [16,20,21].
The purpose of this paper is to contribute to the knowledge of coherent quantum estimation and control by developing further a non-commutative formulation of the quantum filter given by Belavkin in 1980 [1]. While the main results obtained by Belavkin apply only to the commutative measurement case, the problem formulation he used was more general. Building on Belavkin's formulation, we define a non-commutative filtering problem for a class of non-commutative linear systems with non-commutative outputs as a linear least mean square estimation problem. We solve this noncommutative estimation problem within the framework of non-commutative probability theory, expressed as a non-commutative generalization of the Belavkin-Kalman filter. The non-commutative filter is required to be a physical open quantum system, we obtain sufficient conditions for this in terms of algebraic conditions for physical realizability. Generally speaking, the physical realizability imposes some restrictive constraints which makes difficult the use of such non-commutative least mean squares estimators as a physical estimator for many quantum systems described by linear QSDEs.
Furthermore, we explain how Belavkin's quantum filter for the case of commutative outputs (measurements) arises as a special case. A further special case is the Kalman filter for classical linear systems. While applications of our results to coherent quantum feedback control are anticipated, such explorations are beyond the scope of the present paper. This paper is organized as follows. In Section 2, we present general quantum linear stochastic dynamics. In Section 3, we obtain non-commutative linear least mean squares estimators for the general linear quantum stochastic dynamics, expressed in Theorem 2. In Section 4, we study the physical realizability of such linear least mean squares estimators. The main results of this section is expressed in Theorems 3 and 4. In Section 5, we show that our results are in accordance with Belavkin-Kalman filter (commutative outputs) and classical Kalman filter. Finally, the conclusion is given in Section 6.

Quantum linear stochastic dynamics
Consider linear quantum possibly non-commutative stochastic systems of the form [12] Here, A, B, C and D are real matrices in R n×n , R n×nw , R ny×n , and R ny×nw , where n, n w , n y are positive integers with n w ≥ n y . Also, x(t) = [x 1 (t), . . . , x n (t)] T is a vector of self-adjoint possibly non-commutative system variables. In general, the output signal y(t) is non-commutative, and so does not in general correspond to a classical measurement signal. The initial state x(0) is Gaussian with state ρ and satisfy the commutation relation 1 where Θ is a real antisymmetric matrix with components Θ jk and i = √ −1. We assume that the matrix Θ can take one of the two following forms: , with 0 ≤ n ′ ≤ n and n − n ′ even. 1 The notation [A, B] corresponds to AB − BA.
Here J corresponds to the real skew-symmetric 2 × 2 matrix and the "diag" notation corresponds to a block diagonal matrix. Also, diag m (J) denotes a m × m block diagonal matrix with m matrices J on the diagonal. The noise dw(t) is a vector of self-adjoint quantum noises with Itō table where F w is a non-negative Hermitian matrix (see e.g., [24,2]). Indeed, the special case F w = I nw×nw describes a classical noise vector dw. However, the more general case presents n ′ classical noises and n w − n ′ conjugate quantum noises (here I nw×nw is the n w × n w identity matrix). The quantum noises correspond to operators on a Fock space (see e.g., [3,24]). As discussed in [12], we can always set up the following conventions by appropriately enlarging dw, dy, B and C: i) n y is even ii) F w has the following form hence n w should be even.
In general, the system dynamics (1) does not necessarily present any meaningful physical quantum system. In the case that Θ is canonical, the system is physically realizable if it presents an open quantum harmonic oscillator. In quantum mechanics, the canonical commutation relation x(t)x(t) T − (x(t)x(t) T ) T = 2iΘ will be preserved for all t ≥ 0, since the evolution of a closed system in the Heisenberg picture is described by a unitary operator.
In the case that Θ is degenerate canonical, the system is physically realizable if there exists an augmentation of Equation (1) (see more details in [12]) such that the new QSDEs represent the dynamics of an open quantum harmonic oscillator.
The following theorem borrowed from [12] provides necessary and sufficient conditions for physical realizability of Equation (1) for any Θ (canonical or degenerate canonical).
where T w corresponds to the following commutation relation for the noise component dw See the formal definition of physical realizability in [12], where the explicit forms of Hamiltonian and coupling operators are given for both cases, canonical and degenerate canonical Θ. Indeed, the above physical realizability conditions are sufficient for the purpose of this paper.

Linear least mean squares estimation
In this section, we formulate a linear least squares estimation problem for the non-commutative linear system (1). We use the notation Y(t) for the von Neumann algebra generated by the output process y(s) for 0 ≤ s ≤ t. In general, the algebra Y is not commutative.
When Y(t) is commutative, i.e. y(t) is a classical measurement process (by the Spectral Theorem [6, Theorem 3.3]), the optimal filter in the least squares sense is obtained by computing the conditional expectation onto Y(t) [3,6]. The non-demolition property ([x(t), y(s)] = 0, for any t ≥ s) is sufficient to conclude the existence of the commutative conditional expectation [2].
Note that in the present non-commutative case, we seek a dynamical equation for an estimatex(t) ∈ Y(t) driven by the non-commutative output process y(t) defined in the following definition. In contrast to commutative output, it is not shown whether the least mean square estimator that we define in the following is equivalent to conditional expectation. This problem is related to the existence of a non-commutative conditional expectation which is not always guarantied (see more details in [26]). Definition 1. A linear least mean square estimator for the non-commutative linear system (1) is a linear system of the form where K and L are real matrices chosen to ensure for all events Q y ∈ Y t .
Note that Equation (4) has the form of the Kalman filter. Condition We shall refer to the process r(t) defined by as the innovation process. Now let us state the main result of this section through the following theorem.
Theorem 2. Let x be quantum Gaussian variable satisfying the noncommutative linear system (1). Now consider the estimator of the form (4) with the following assumptions.
(i) Take the initial condition this means that the estimator has canonical commutation relation initially; (ii) Suppose that the initial symmetric covariance matrix is given by (iii) the plant has canonical commutation relation initially, i.e., The notation X * corresponds to the adjoint of the operator X.
(iv) x(0) and x(0) are initially decoupled, i.e., Then, a system of the form (4) is a linear least mean square estimator if and only if where the real symmetric matrix P defined as satisfies the following Riccati equatioṅ and the real antisymmetric matrix Q defined as satisfies the following equatioṅ Before giving the proof of Theorem 2, we first prove the following essential lemma which is what we need to prove Theorem 2.
. Then, we can express g τ (t) as follows Then, we find Hence, we get Now we can state the proof of Theorem 2.
Proof of Theorem 2. Assume that the system (4) is a linear least mean square estimator for the system (1). Then, setting Q y = I in the orthogonality condition (5) implies L(t) = A.
The error as defined in Equation (6) satisfies the following dynamics Since (4) is a linear least mean square estimator, by setting Q y = r(τ ) (componentwise) in (5), we get the following equality for all t ≥ τ. Now apply Lemma 1, Equality (11) is true if and only if Differentiation of Equation (12) gives We can conclude E ρ de (τ ) r (τ ) T = 0 because of (11) and the definition of Itō process. Therefore, we find where we have used Equations (7) and (10). We point out that By Equation (13), we have the following equality By definitions of P and Q given in the theorem, we get obviously the following equality E ρ e (τ ) e (τ ) T = P (τ ) + iQ(τ ).
Equation (14) is equivalent to Therefore, we find As all the matrices in Equation (16) are real, we have the following equalities This leads to the following equalities With a little computation (writing the derivation of E ρ [ee T ]), we find the Riccati equations for P and Q given in Equations (8) and (9) respectively.
Conversely, by contradiction, suppose that the linear least mean square estimator is given by the following where at least one the following inequalities holds Then, it is sufficient to show that x cannot satisfy the orthogonality condition given in Equation (5) for any Q y . In order to show this, it is sufficient to suppose that Equality (5) holds for any Q y , then by taking Q y = I and Q y = r(τ ), we can easily show the contradiction which follows.

Physical realizability
In this section, we will study the physical realizability of the least mean squares estimators announced in Theorem 2. In Theorem 2, we don't assume either the system (1) nor the linear least mean square estimator (4) to be physically realizable. If we impose the physical realizability conditions of the system (1), we get two following equalities which are obtained from Equation

General results on physical realizability
In this subsection, we will announce a general theorem which gives necessary and sufficient conditions ensuring physical realizability of the least mean squares estimators given in Theorem 2. Proof. If we impose the physical realizability constraints on the estimator of the form (4), Q takes the following form (see more details in [12, Theo- where for the last equality, we suppose that the plant and the estimator are decoupled at any t ≥ 0. Therefore, from Equation (20), we get the following If we multiply the above equation by diag ny 2 (J), we find where for the last equality, we have used relation (3).
As a result, we take the following expression for K Hence, the linear least mean square estimator becomes In order to make the above estimator physical, we require that Now using the physical realizability of the plant given in Equation (3), we find that with Θ = J and F = I 4×4 + idiag 2 (J). It can be easily verified that such a plant is physically realizable.
By Theorem 3, we find With a little computation, we get the following estimator We can verify that such a linear estimator is physical if and only if 0 3κ 1 −3κ 1 0 = 0 which is equivalent to κ 1 = 0. Note that this condition is conform with the second condition given in Theorem 3, since by replacing B by its value and n w = 4 and n y = 2, we find

Special results on physical realizability
Let us write B as B = [B ′ n×ny B ′′ n×nw−ny ]. In the following, first, we study the physical realizability of the least mean square estimator announced in Theorem 2 for the case where B ′ diag ny 2 (J)B ′T = 0. Second, we consider the special case n w = n y . Some illustrative examples will be provided.
As it is demonstrated in the following theorem, the physical realizability constraints impose some restrictive conditions on B depending on the forms of Θ and C. (ii) Take Θ canonical, then, K = 0, (equivalently, B ′ = 0); (iii) Take Θ degenerate canonical, then,

Proof. Condition (ii) in Theorem 3 is equivalent to the following
Hence, the condition (i) is immediate. In the following, we prove the validity of condition (iii). The condition (ii) can be shown in the same way.
Then, take Θ degenerate canonical. By Equation (21), we have If we multiply Equation (24) by DB T , we find As a result, if B ′ diag ny 2 (J)B ′T = 0, equivalently, BD T diag ny 2 (J)DB T = 0, then we have necessarily the following which is equivalent to B ′ = 0.

Example 2. Consider the following plant
with Θ = J and F = I 4×4 + idiag 2 (J). It can be easily verified that such a plant is physically realizable.
It is trivial that the condition B ′ JB ′T = 0 is satisfied. By Theorem 4, a least mean square estimator which is physical should satisfy K = −B ′ = 0, i.e., κ 2 = 0. This can be also obtained by writing the least mean square estimator proposed by Theorem 3, i.e., which is physical if and only if κ 2 = 0.
Case 2 : n y = n w . In this case, a physical realizable plant should satisfy D = I ny×ny . As a result, the plant given in (1) takes the following form For this special case, the results of Theorem 3 are simplified to (ii) for Θ degenerate caonical: As a result, we can observe that for the case n y = n w , it is often impossible to construct a physical realizable least mean square estimator of the form (4).

Example 3. Consider the following plant
with Θ = diag(0, J) and F w = I 4×4 + idiag 2 (J). It can be easily verified that such a plant is physically realizable. In addition, Bdiag 2 (J)B T = 0 is satisfied.
by applying Corollary 1, we take K = −B, we have (26) With a little computation, we can easily check that such an estimator is physically realizable which is also conform with Corollary 1.

Example 4. Consider the following plant
with Θ = diag(0, J) and F w = I 4×4 + idiag 2 (J). The matrices A, and B have the same forms as the ones appeared in previous example. However, C satisfies diag 0 n ′ ×n ′ , diag n−n ′ 2 (I) C T = C T . Then, by Corollary 1, the following physically realizable estimator is not the least mean square estimator unless κ 2 = 0 and η = 0, i.e., K = 0.
Example 5. Consider an optical cavity which is of the form where dw(t)dw(t) T = (I 2×2 + iJ) dt and Θ = J.
If we take K = −B, we find Such an estimator is physical if and only if κ = 0 which is also conform with Corollary 1.

Consistency with standard results
In this section, we show that the least mean squares estimators found in Theorem 2 are reduced to Belavkin-Kalman filters [10] under the assumptions that Belavkin used, i.e., the commutativity of the outputs and the non-demolition property. Also, we discuss the classical Kalman filters as a special case of Theorem 2.

Non-commutative dynamics, commutative (classical) outputs
In this subsection, we take Y t to be commutative, that is y(t) is self-adjoint for each t and [y(t), y(s)] = 0 for all s, t. By the Spectral Theorem, [6,Theorem 3.3]), y(t) corresponds to a classical stochastic process, the measurement process. Such continuous measurement signal arise in Homodyne detection [30].
For the commutative outputs, we have the following correlation for the observation processes dy dy(t)dy T (t) = F y dt, with F y = I ny×ny . Note that we have the following relation between F w and F y , DF w D T = F y , with D = [I ny×ny 0 ny×(nw−ny) ].
Therefore, F w takes the following form It is well known that the optimal filter satisfies the following dynamics with x(0) = E[x(0)] =x(0), and Remark that the variable x has zero commutation relation for any t ≥ 0, i.e., this means that x is a classical variable.
The real symmetric matrix P (t) := 1 2 E e (t) e (t) T + (e (t) e (t) T ) T satisfies the following Riccati equatioṅ For commutative output, we find Q = Θ by using Equation (29) and the following commutation relations Now let us show that the dynamics of the optimal filter (28) can be obtained by Theorem 2.
By replacing F w with its value given in (27) into Equation (14), we find Then, we get Therefore, we find The second equation in (30) is always satisfied, as by the non-demolition property, we have ΘC T + BT w D T = 0. Hence, we get ΘC T = 0.

Classical systems
The classical linear stochastic dynamics is described by classical variables as follows where A, B, C and D are real matrices in R n×n , R n×nw and R ny×n and R ny×nw , and dw is a classical Wiener process, with dw(t)dw(t) T = I nw×nw dt.
Take Y(t) as the algebra generated by the observation processes previous to time t, defined by Y(t) := span{(y(s)) 0≤s≤t }.
It is well known that the classical Kalman filter [25,17] satisfies the following dynamics Note that the Kalman filter can be obtained directly from Theorem 2 with taking F w = I nw×nw , F y = I ny×ny and Θ = 0.

Conclusion
We have obtained non-commutative linear least mean squares estimators for linear QSDEs by extending Belavkin-Kalman filters to the case where the output processes are non-commutative. We have assumed that these least mean squares estimators are given as a linear combination of innovation processes. Furthermore, we studied the physical realizability of such estimators for general case and some special cases. We have shown that such linear least mean square estimator becomes a physical realizable least mean square estimator if and only if When Θ has degenerate canonical form, if C has a particular form, then K = −B = 0. Also, for canonical case, we find K = −B = 0.
The results appeared in this paper demonstrates that the non-commutative linear least mean squares estimators cannot be realized physically in most of the time, especially when n y = n w .
Also, we have shown that our results are in agreement with standard results, i.e, Belavkin-Kalman and classical Kalman filtering.
Note that this work does not show that the best estimate based on the knowledge of the non-commutative output processes, has the form of the proposed linear estimator (4). Our future aim is to solve the optimal filtering problem under the non-convex constraints imposed by physical realizability conditions. Furthermore, the optimal filtering problem when the coherent controllers are added into the plant's dynamics (see e.g., [23]) are in our future research plan.