On the generalization of linear least mean squares estimation to quantum systems with noncommutative outputs
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Abstract
The purpose of this paper is to study the problem of generalizing the BelavkinKalman filter to the case where the classical measurement signal is replaced by a fully quantum noncommutative output signal. We formulate a least mean squares estimation problem that involves a noncommutative system as the filter processing the noncommutative output signal. We solve this estimation problem within the framework of noncommutative probability. Also, we find the necessary and sufficient conditions which make these noncommutative estimators physically realizable. These conditions are restrictive in practice.
Keywords
linear quantum stochastic differential equations (QSDEs) quantum noises Kalman filtering physical realizability linear least mean squares estimators noncommutative outputs coherent observers1 Introduction
Quantum filtering theory as a fundamental theory in quantum optics, which was implicit in the work of Davies in the 1960s [1, 2] concerning open quantum systems and generalized measurement theory, and culminating in the general theory developed and initiated by Belavkin during the 1980s [3, 4, 5]. 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 [6, 7].
One application of the quantum filter, or variants of it, is in measurement feedback control [7, 8, 9, 10, 11, 12, 13]. 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 [10, 14]. However, feedback control of quantum systems need not involve measurements, and indeed the topic of coherent quantum feedback is evolving [14, 15, 16, 17, 18, 19, 20], 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 recently led to new proposals for quantum memories, quantum error correction, and ultralow power classical photonic signal processing [21, 22, 23, 24].
The purpose of this paper is to contribute to the knowledge of coherent quantum estimation and control by developing further a noncommutative formulation of the quantum filter given by Belavkin in 1980s [3]. While the main results obtained by Belavkin apply only to the commutative measurement case, the problem formulation he used was more general. Belavkin’s theory of quantum filtering concerns the estimation of the variables of quantum systems conditioned on classical (commutative) measurement records. For linear quantum stochastic systems, Belavkin’s filter has the same form as the classical Kalman filter. The BelavkinKalman filter is a classical system that processes the incoming measurements to produce the desired estimates. The estimates may be used for monitoring and/or feedback control of the quantum system.
In our study, we formulate and solve a problem of optimal estimation of a linear quantum system variables given the noncommutative outputs within the framework of noncommutative probability theory. In particular, we derive a system of noncommutative stochastic differential equations (the noncommutative BelavkinKalman filter) that minimizes a least squares error criterion. Such noncommutative filtering equations are well defined mathematically, even if they do not correspond to a physical system. However, if we wish to implement the noncommutative BelavkinKalman filter within the class of physically realizable linear quantum stochastic systems (such as linear quantum optical systems), then we find that the conditions for physical realizability impose strong restrictions. In this paper, we find physical realizability conditions for general case and also for some particular cases. These strong physical realizability conditions are a key contribution of this paper.
We remark that our contribution here is different from the problem studied in [25]. Since, in [25], the authors propose another physically realizable quantum system considered as a filter, connected to the output of the plant whose dynamics can be determined by minimizing the mean square discrepancy between the plant’s state and the output of the filter. Also, they suppose an additional vacuum noise other than the plant’s noises in the form of filter’s dynamics. However, in this paper, we focus firstly on finding the form of linear least mean squares estimators for the noncommutative outputs by temporarily excluding the physical realizability constraints. To do this, we proceed as classical Kalman filtering and BelavkinKalman filtering by supposing that the mean squares estimator should satisfy a linear dynamics of innovation processes and we do not suppose any additional vacuum noises other than dw which is the input process of the plant. As such, we obtain the form of least mean squares estimators for non commutative outputs. Then, we seek necessary and sufficient conditions which make such linear least mean squares estimators automatically physically realizable. As we can observe in examples, for some particular forms of plants, we are obliged to add additional vacuum noises to the least mean squares estimators to ensure physical realizability. These estimators which track asymptotically the plant’s state and are physically realizable are called coherent observers [26]. Roughly speaking, coherent least mean squares estimators and observers are another physical systems connected to the main system in cascade [15, 26, 27]. We remark that coherent linear least squares estimators and observers could in principle be used for coherent feedback control, although this matter is outside 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 noncommutative 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 are expressed in Theorem 3 and Corollaries 16. Moreover, some illustrative examples are provided. Finally, the conclusion is given in Section 5.
2 Quantum linear stochastic dynamics

Canonical if \(\Theta=\operatorname{diag}_{\frac{n}{2}}(J)\), with n even or

Degenerate canonical if \(\Theta=\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac{nn'}{2}}(J) )\), with \(0< n'\leq n\) and \(nn'\) even.^{2}
2.1 Physical realizability of linear QSDEs
Not all QSDEs of the form (1) represent the dynamics of physically meaningful open quantum systems. In the case that Θ is canonical, the system is physically realizable if it presents an open quantum harmonic oscillator. Now we give the formal definition of physical realizability (see e.g., [18], Definition 3.3).
Definition 1
The following theorem borrowed from [18] provides necessary and sufficient conditions for physical realizability of Equation (1) for any Θ (canonical or degenerate canonical).
Theorem 1
([18])
In the following lemma, we prove that the nondemolition property holds for noncommutative outputs if system (1) is physically realizable.
Lemma 1
Proof
In the following lemma, we show that the noncommutative outputs do not have selfnondemolition property.
Lemma 2
Proof
Remark 1
Before starting the next section, let us present the following definition.
Definition 2
3 Linear least mean squares estimation
Take \(\mathcal{Y}(t)\) for the von Neumann algebra generated by the output process \(y(s)\) for \(0\leq s \leq t\). When \(\mathcal{Y}(t)\) is commutative, i.e., \(y(t)\) is a classical measurement process (by the spectral theorem [32], Theorem 3.3), the optimal filter in the least squares sense is obtained by computing the conditional expectation onto \(\mathcal{Y}(t)\) [5, 32]. The nondemolition property (\([x(t), y(s)^{T}] = 0\), for any \(t \geq s\)) is sufficient to conclude the existence of the commutative conditional expectation [29].
In contrast to commutative output, it is not shown whether the least mean squares estimator that we define in Definition 3, is equivalent to conditional expectation. This problem is related to the existence of a noncommutative conditional expectation which is not always guaranteed, and we do not consider this matter in this paper (see more details in [33]).
Definition 3

it is defined on the class ξ, i.e., it is a linear system of the form (10), and
 the real matrix \(K(t)\) is chosen to minimize the symmetrized mean squares error defined as followswhere \(P(t)\) is the symmetric error covariance matrix defined by$$ J\bigl(K(t)\bigr):=\operatorname{Tr}\bigl[P(t)\bigr], $$(11)$$ P(t):=C_{e(t)}=\frac{1}{2} \mathbb{E}_{\rho}\bigl[ e(t) e(t)^{T}+\bigl(e(t)e(t)^{T}\bigr)^{T} \bigr]. $$
Theorem 2
Proof
4 Results on 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 do not assume the linear least mean squares estimator (10) to be physically realizable.
Theorem 3
Proof
Moreover, Equation (27) is derived from the Riccati equation (19) by replacing K by its value given in Equation (13) and using the physical realizability of the plant. □
4.1 Some special cases
In the following, first, we study the physical realizability of the least mean squares estimator announced in Theorem 2 for the case where \(B'\operatorname{diag}_{\frac{n_{y}}{2}}(J) B^{\prime T}=0\), with \(B'\) defined in Equation (25). Second, we study the case where \(B'=0\).
4.1.1 Case 1: \(\mathbf{B}'\operatorname{diag}_{\frac{n_{y}}{2}}(J) B^{\prime T}=0\)
As it is demonstrated in the following corollary, the physical realizability constraint announced in (26) can be simplified.
Corollary 1
 (i)
\(K=B'+PC^{T}\).
 (ii)For Θ canonical,with P the unique symmetric positive definite solution of the following Riccati equation,$$ B''\operatorname{diag}_{\frac{n_{w}n_{y}}{2}}(J)B^{\prime\prime T}+2P \Theta B'B^{\prime T}+2B'B^{\prime T} \Theta P=0, $$(29)$$ \begin{aligned} &\dot{P}(t) =AP(t)+P(t)A^{T}+P\Theta B'B^{\prime T}\Theta P(t)+B''B^{\prime\prime T}, \\ &P(0) =C_{e(0)}. \end{aligned} $$(30)
 (iii)For Θ degenerate canonical,
 (i)if \(\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac {nn'}{2}}(I) )C^{T}=C^{T}\), thenwith P satisfying dynamics (30),$$ B''\operatorname{diag}_{\frac{n_{w}n_{y}}{2}}(J)B^{\prime\prime T}+2P \Theta B'B^{\prime T}+2B'B^{\prime T} \Theta P=0, $$
 (ii)but if \(\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac{nn'}{2}}(I) )C^{T}\neq C^{T}\) holds, then,with P satisfying Equation (27).$$\begin{aligned}& B''\operatorname{diag}_{\frac{n_{w}n_{y}}{2}}(J)B^{\prime\prime T}+2PC^{T} \operatorname{diag}_{\frac {n_{y}}{2}}(J)B^{\prime T} \\& \quad{}+2B'\operatorname{diag}_{\frac{n_{y}}{2}}(J)CP+PC^{T} \operatorname{diag}_{\frac {n_{y}}{2}}(J)CP=0, \end{aligned}$$(31)
 (i)
Proof
Now we consider the case Θ degenerate canonical, in this case if we multiply the expression of CΘ (and similarly, \(\Theta C^{T}\)) given in above by Θ, we get \(C\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac {nn'}{2}}(I) )=\operatorname{diag}_{\frac{n_{y}}{2}}(J)DB^{T}\Theta\) (and similarly, \(\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac {nn'}{2}}(I) )C^{T}=\Theta BD^{T}\operatorname{diag}_{\frac{n_{y}}{2}}(J)\)). Now it is clear that if the condition \(\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac{nn'}{2}}(I) )C^{T}=C^{T}\) holds, then we get exactly the constraint given in the first part. However, if this condition does not hold, from (26), we get the constraint (32) which is exactly the condition (31). □
Remark 2
We remark that the condition \(B'\operatorname{diag}_{\frac{n_{y}}{2}}(J) B^{\prime T}=0\) was considered in order to simplify the physical realizability constraints in Equation (26). As the corollary in above shows, in most of the times, this case is equivalent to eliminating the quadratic terms in Equation (26). Also, note that if \(n_{y}=n=2\), the condition \(B'\operatorname{diag}_{\frac{n_{y}}{2}}(J) B^{\prime T}=0\) is equivalent to the condition \(\det(B')=0\), i.e., the quadratures are linearly dependent.
Particular case: \(n_{y}=n_{w}\)
Corollary 2
 (i)
\(K=B+PC^{T}\).
 (ii)For Θ canonical,with P satisfying the following Riccati equation$$ 2P\Theta BB^{T}+2BB^{T}\Theta P=0, $$(34)$$ \begin{aligned} & \dot{P}(t) =AP(t)+P(t)A^{T}+P(t)\Theta BB^{T} \Theta P(t), \\ &P(0) =C_{e(0)}. \end{aligned} $$(35)
 (iii)For Θ degenerate canonical,
 (i)if \(\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac {nn'}{2}}(I) )C^{T}=C^{T}\), thenwith P satisfying dynamics (35),$$ 2P\Theta BB^{T}+2BB^{T}\Theta P=0, $$
 (ii)but if \(\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac{nn'}{2}}(I) )C^{T}\neq C^{T}\) holds, then,with P satisfying the following dynamics$$\begin{aligned}& 2PC^{T}\operatorname{diag}_{\frac{n_{y}}{2}}(J)B^{T} +2B\operatorname{diag}_{\frac{n_{y}}{2}}(J)CP \\& \quad{}+PC^{T}\operatorname{diag}_{\frac{n_{y}}{2}}(J)CP=0, \end{aligned}$$(36)$$ \begin{aligned} &\dot{P}(t) =(ABC)P(t)+P(t) (ABC)^{T}P(t) \bigl(C^{T}C\bigr)P(t), \\ &P(0) =C_{e(0)}. \end{aligned} $$(37)
 (i)
Proof
The proof of this corollary can be done by the same arguments provided for Corollary 1. Since, if \(n_{y}=n_{w}\), the condition \(B'\operatorname{diag}_{\frac{n_{y}}{2}}(J) B^{\prime T}=0\) is equivalent to \(B\operatorname{diag}_{\frac{n_{w}}{2}}(J) B^{T}=0\). □
Particular case: \(n=2\), \(n_{y}=2\), \(n_{w}=4\), and \(\Theta=J\)
Consider the simple case \(n=2\), \(n_{y}=2\), \(n_{w}=4\), and \(\Theta=J\). Take \(A= \bigl({\scriptsize\begin{matrix}a_{1}&a_{2}\cr a_{3}&a_{4} \end{matrix}}\bigr) \), \(P= \bigl({\scriptsize\begin{matrix}p_{1}&p_{2}\cr p_{2}&p_{4} \end{matrix}}\bigr) \), \(B'= \bigl({\scriptsize\begin{matrix}b_{1}&b_{2}\cr b_{3}&b_{4} \end{matrix}}\bigr) \), and \(B''= \bigl({\scriptsize\begin{matrix}d_{1}&d_{2}\cr d_{3}&d_{4} \end{matrix}}\bigr) \). In the following, we find the constraints which guarantee the physical realizability of the least mean squares estimator announced in Theorem 2.
Corollary 3
Proof
The proof can be directly derived from Equation (29). □
Now, we can conclude the following corollary.
Corollary 4
Suppose \(b_{1}=b_{3}\), \(b_{2}=b_{4}\), and \(\det(B'')=0\). Then, the linear least mean squares estimator announced in Theorem 2, is physically realizable if and only if \(p_{1}+p_{4}=2p_{2}\).
The following corollary shows the difficulty of finding a physical realizable least mean squares estimator for some particular forms of P, \(B'\) and \(B''\).
Corollary 5
Suppose \(b_{1}=b_{3}\), \(b_{2}=b_{4}\), \(d_{1}=d_{3}\), and \(d_{2}=d_{4}\). Then, it is impossible to realize physically a linear least mean squares estimator of the form given in (10) such that \(p_{1}=p_{2}=p_{4}\).
Proof
By Corollary 4, we know that if \(b_{1}=b_{3}\), \(b_{2}=b_{4}\), and \(\det(B'')=0\), then the physical realizability condition (38) implies that \(p_{1}+p_{4}=2p_{2}\). Thus, when \(p_{1}=p_{2}=p_{4}\), this condition is satisfied.
This result shows that in order to obtain conditions on B, which make the linear least mean squares estimator given in Theorem 2 physically realizable (for e.g., see Equation (38)), we need to suppose some constraints on P. This demonstrates the difficulty to find an appropriate plant whose least mean squares estimator is physically realizable.
4.1.2 Case 2: \(\mathbf{ B}'=0\)
Let us announce the following corollary for this special case.
Corollary 6
 (I)
For canonical Θ, we have \(K=0\), and \(B''\operatorname{diag}_{\frac{n_{w}n_{y}}{2}}(J)B^{\prime\prime T}=0\);
 (II)For degenerate canonical \(\Theta=\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac{nn'}{2}}(J) )\), we have
 (i)
If \(\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac {nn'}{2}}(I) )C^{T}=C^{T}\), then \(K=0\), and \(B''\operatorname{diag}_{\frac{n_{w}n_{y}}{2}}(J)B^{\prime\prime T}=0\).
 (ii)If \(\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac {nn'}{2}}(I) )C^{T}\neq C^{T}\), then \(K=PC^{T}\), and \(B''\operatorname{diag}_{\frac{n_{w}n_{y}}{2}}(J)B^{\prime\prime T}+PC^{T}\operatorname{diag}_{\frac {n_{y}}{2}}(J)CP=0\), with P satisfying$$ \begin{aligned} &\dot{P}(t) =AP(t)+P(t)A^{T}P(t) \bigl(C^{T}C \bigr)P(t)+B''B^{\prime\prime T}, \\ &P(0) =C_{e(0)}. \end{aligned} $$(41)
 (iii)Let us write \(C=[ C'_{n_{y}\times n'} \ C''_{n_{y}\times (nn')} ]\). Then, ifand \(C^{\prime T}\operatorname{diag}_{\frac{n_{y}}{2}}(J)C'=0\), then \(K=PC^{T}\), and \(B''\operatorname{diag}_{\frac{n_{w}n_{y}}{2}}(J)B^{\prime\prime T}=0\), with P satisfying the Riccati equation (41).$$\operatorname{diag} \bigl(0_{n'\times n'},\operatorname{diag}_{\frac {nn'}{2}}(I) \bigr)C^{T}\neq C^{T} $$
 (i)
Proof
If Θ is canonical, then we find \(K=0\), since \(B'=0\) implies \(C=0\) by physical realizability conditions given in Theorem 1 (Equation (6)). Then, we can use the results of Corollary 1, where by replacing \(B'=0\) in Equations (29) and (30), we find the conditions given in part (I).
However, if Θ is degenerate canonical, C is not necessarily zero if \(C\Theta=0\). In this case, we find \(K=PC^{T}\). If \(\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac {nn'}{2}}(I) )C^{T}= C^{T}\), then \(C=0\). This proves the results given in (i) of part (II).
But if \(\operatorname{diag} (0_{n'\times n'},\operatorname{diag}_{\frac {nn'}{2}}(I) )C^{T}\neq C^{T}\), then we have to replace \(B'=0\) in Equations (31) and (27), but C is not necessarily zero. This proves the conditions (ii) of part (II). The condition (iii) in part (II) can be derived from condition (ii). Also, by noting that \(C\Theta=0\), implies \(C''=0\). □
Note that \(B'=0\) implies \(C\Theta=0\). Roughly speaking, when \(C\Theta =0\), the noncommutative filter obtained in the above theorem, could also be realized with Homodyne or Hetrodyne detection. Since, no quantum information is transferred from the plant to the filter in this case. This is like the classical filtering cases of Homodyne or Heterodyne detection, where one always ends up taking a single quadrature of the field.
4.2 Consistency with standard results
In this subsection, we recall the standard results, i.e., BelavkinKalman and classical Kalman filtering. They can be respectively considered as special cases when the output is commutative but the plant’s dynamics is noncommutative and when the output and the plant’s dynamics are both commutative.
Noncommutative dynamics, commutative (classical) outputs
It can easily be shown that the least mean squares estimators found in Theorem 2 are reduced to BelavkinKalman filters [9] under the assumptions that Belavkin used, i.e., the commutativity of the outputs and the nondemolition property.
Take \(\mathcal{Y}_{t}\) to be commutative, that is \(y(t)\) is selfadjoint for each t and \([y(t), y(s)^{T}]=0\) for all s, t. By the spectral theorem, [32], Theorem 3.3, \(y(t)\) corresponds to a classical stochastic process, the measurement process. Such continuous measurement signal arise in Homodyne detection [7].
Classical Kalman filtering
4.3 Construction of coherent observers with least mean squares estimators
4.4 Examples
In the following, we give some examples from the literature to illustrate the results of this section. Also, these examples show the difficulty to find an example where construction of a physically realizable least mean squares estimator is feasible.
Example 1
Example 2
If we take \(B'=0\) (with previous notation), i.e., \(\kappa_{1}=0\), the physically realizable constraint (49) is satisfied if and only if \(\kappa_{2}=0\). This means that both field channels are decoupled from the system. Moreover, if we take \(\kappa_{1}=\kappa_{2}=0\), the Riccati equation (50) has not a unique solution. Also, for \(\kappa_{2}\geq0\) and \(\kappa_{1}>0\), the physical realizability condition given in (49) imposes a constraint on the form of P. This shows the restrictiveness of physical realizability constraints.
Example 3
Now take the following parameters: \(\kappa=0.1\), \(\epsilon_{r}=0.01\), and \(\epsilon_{i}=0.01\). In this case, we get \(P= \bigl({\scriptsize\begin{matrix} 1.2000 & 0.2000\cr 0.2000 & 0.8000 \end{matrix}}\bigr) \) and \(K= \bigl({\scriptsize\begin{matrix} 0.0632& 0.0632\cr 0.0632& 0.0632 \end{matrix}}\bigr) \). The performance of the least mean squares estimator is given by \(J(K)=\operatorname{Tr}(P)=2\).
Example 4
So if \(\kappa_{2}=0\), the physical realizability condition (53) is satisfied. This means that the system should be decoupled from the field channel \(dw_{1}\). However, if \(\kappa_{2}=0\), for \(\Delta \neq0\), the Riccati equation (54) has no solution if \(\kappa_{3}\neq0\). Moreover, the Riccati equation (54) has not a unique solution if \(\kappa_{2}=\kappa_{3}=0\). Also, if \(p_{1}=0\), the condition (53) is satisfied. However, it is not difficult to show that there is no positive definite solution to the Riccati equation above in this case. This illustrates once again the restrictive nature of the physical realizability conditions.
For \(\tilde{x}\), we find \(\tilde{P}= \bigl({\scriptsize\begin{matrix} 0.7075& 1.1760\cr 1.1760 & 33.9878 \end{matrix}}\bigr) \), then \(\tilde{J}(K)=\operatorname{Tr} (\tilde{ P})=34.6953\).
Example 5
We have observed in the examples above, constructing physically realizable least mean squares estimators was impossible when \(B'\neq0\) or we should consider some constraints on the matrix P which makes the problem hard and sometimes impossible to solve. This shows the restrictiveness of the physical realizability constraints. Also, when \(B'=0\), the physically realizable least mean squares estimators are not well defined. Supported by these examples and some others which are not given in this paper, we conclude that maybe it is impossible to find examples which could result in physically realizable least mean squares estimators without any additional quantum noises when \(B'\neq0\). (Note that the case \(B'=0\) is not an interesting case, since it could also be realized with Homodyne or Hetrodyne detection, as mentioned before, below Corollary 6.) However, we could not show this in general case, maybe it is wrong. Also, note that finding examples is a hard problem since we should solve the quadratic equations in P (Equation (27)) where we obtain P as a function of free parameters of the matrix A, and B. Then, these free parameters could be determined by replacing P in the physical realizability constraints (Equation (26)).
5 Conclusion
We have obtained noncommutative linear least mean squares estimators for linear QSDEs by extending BelavkinKalman filters to the case where the output processes are noncommutative. 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 the general case and some special cases.
We have observed that when \(B'=0\), it is more simple to construct a physically realizable least mean squares estimator, specially for Θ degenerate canonical and when \(C^{T}\operatorname{diag}_{\frac {n_{y}}{2}}(J)C=0\). Since, in this case, the physical realizability condition does not depend on the form of P (see more details in Corollary 6). However, roughly speaking, for this case, the noncommutative filter could also be realized by Homodyne or Hetrodyne detection as \(C\Theta=0\). In general, finding examples which satisfy physical realizability conditions, it is difficult without any assumptions on P. These assumptions create constraints on their associated Riccati equations (see e.g., Theorem 3 and Corollaries 16). Moreover, based on our observations, we can conclude that maybe, the construction of a physically realizable least mean squares estimator without any additional quantum noises is impossible when \(B'\neq0\). Generally speaking, the results presented here show the restrictive nature of physical realizability conditions.
Indeed, this work does not show that the best estimate based on the knowledge of the noncommutative output processes, and under the constraints of the physical realizability, has the form of the proposed linear estimator (10). Further research is required to solve the optimal filtering problem under the nonconvex 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., [15, 41]) can be considered as a future research plan.
Footnotes
 1.
The notation \([A,B]\) corresponds to \(ABBA\).
 2.
Here \(0_{m\times n}\) corresponds to \(m\times n\) zero matrix.
 3.
The notation \(X^{T}\) corresponds to the transpose of the matrix X.
 4.
Here \(I_{n\times n}\) is the \(n\times n\) identity matrix.
 5.
If X and Y are column vectors of operators, the commutator is defined by \([X,Y^{T}]=XY^{T}(YX^{T})^{T}\).
Notes
Acknowledgements
The authors gratefully acknowledge Professor T. Duncan for helpful discussions. This work was supported by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (Project No. CE110001027) and Air Force Office of Scientific Research (AFOSR) under Grant No. FA23861214075. The first author would like to thank Ryan Hamerly and Nikolas Tezak for valuable discussions. Nina H. Amini acknowledges the support of Math+X Postdoctoral Fellowship from the Simons Foundation.
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