Design of coherent quantum observers for linear quantum systems

Quantum versions of control problems are often more difficult than their classical counterparts because of the additional constraints imposed by quantum dynamics. For example, the quantum LQG and quantum H infinity optimal control problems remain open. To make further progress, new, systematic and tractable methods need to be developed. This paper gives three algorithms for designing coherent observers, i.e., quantum systems that are connected to a quantum plant and their outputs provide information about the internal state of the plant. Importantly, coherent observers avoid measurements of the plant outputs. We compare our coherent observers with a classical (measurement-based) observer by way of an example involving an optical cavity with thermal and vacuum noises as inputs.


Introduction
Feedback control of quantum systems can be broadly categorized into two schemes: 'classical' (or measurement-based) and 'coherent' control. Classical control involves making projective measurements on the plant (for example homodyne or heterodyne detection in the case of optical systems) and then generating feedback control signals based on these measurements. For a treatment of this topic see for example [1]. In this paper, we are concerned with coherent control which uses controllers that are themselves quantum systems, coupled directly to the plant. One advantage of coherent control schemes is that they avoid the loss of quantum information that occurs during measurement. Coherent quantum control is an active research area [2][3][4][5][6][7][8][9][10] and recent results [10] indicate regimes in which coherent controllers perform better than the optimal classical controllers. Coherent observers represent an important building block in developing systematic and tractable approaches to coherent control problems.
Feedback control schemes require that the controller have access to information about the internal state of the plant. For example, in the classical LQG problem, the Kalman filter [16,17] is used to obtain an optimal estimate of the plant's internal state from a series of noisy measurements. In coherent control schemes, the controller does not have access to measurements, rather it must make use of information from its direct coupling with the plant. A coherent observer is a quantum system that is designed such that when directly coupled with a plant, its outputs provide information about the internal state of the plant. Different approaches to design coherent observers are discussed in [18,19]. However, as with the quantum control problems mentioned above, the main difficulties come from the fact that a coherent observer should satisfy physical realizability constraints.
The main contribution of this paper is to extend previous physical realizability results [13][14][15] to obtain algorithms for designing coherent observers. The previous results we use demonstrate how strictly proper, linear time invariant (LTI) systems can be made physically realizable by allowing additional quantum noises. We extend these results by proving that when a Kalman filter is modified by adding quantum noises as prescribed in [13][14][15], the result is a coherent observer. We give three different algorithms for designing coherent observers in this way. We also give calculations for obtaining a performance metric to compare the coherent observers. Algorithms 2 and 3 incorporate novel approaches to improving performance.
We now outline our three algorithms. The first algorithm is based on the Kalman filter which is modified by allowing additional vacuum noise inputs such that the resulting system is physically realizable. The second algorithm attempts to improve on the first by incorporating a free parameter over which we optimize. The purpose of this parameter is to compensate for the effect of the additional quantum vacuum noises. The third algorithm attempts to find a state transformation of the Kalman filter such that it can be made physically realizable with the minimal number of additional quantum noises. Despite being suboptimal estimations, these algorithms provide a systematic and tractable approach to coherent observer design. Like the celebrated Kalman filter, it is envisaged that the coherent observer will play an important role in the solution to coherent control problems.
The remainder of this paper is structured as follows. In Section 2, we introduce models to describe the quantum systems. In Section 3, we formally define Physical Realizability and present relevant existing results which we will utilize. We are then able to give our problem formulation in Section 4. Section 5 contains the main contribution of the paper: we present three algorithms for designing a coherent observer. In Section 6, we present a measurement-based (classical) observer. The three algorithms are then compared with each other and the measurement-based alternative by way of an example in Section 7. Finally, we give our conclusion in Section 8.

Linear Quantum System Models
In this section, we introduce the linear quantum stochastic models that we will consider for the plants' dynamics. Also, we present the class of coherent observers that we will propose.
As in [2,3,18,19], we consider quantum plants described by quantum stochastic differential equations (QSDEs) of the following form: Here x(t), dy(t), dw 1 (t) and dw 2 (t) are column vectors of operators with even dimensions n x ,n y ,n w 1 and n w 2 respectively. x(t) represents the internal state of the quantum plant, while dw 1 (t) and dw 2 (t) represent the input field(s) and dy(t) represents the output field(s). For simplicity, we restrict our attention to the case where n x = n y . Also, for notational convenience, we separate the input fields arbitrarily such that n w 1 = n x and n w 2 ≥ 0. Being quantum in nature, the state variables x(t) satisfy commutation relations described by a real skew symmetric matrix Θ: We restrict our attention to plants driven by noise processes dw 1 (t) and dw 2 (t) which are quantum Wiener processes with Ito relations dw i (t) dw i (t) T = F w i dt. Here F w i is Hermitian and F w i = S w i + iT w i , where S w i and T w i are the real and imaginary parts of F w i , respectively. The commutation relations for dw i (t) are determined by T w i while the intensity of the noise processes is described by S w i . In particular, we will consider thermal noises with where k n is a parameter describing the intensity of the thermal noise. The case where k n = 0 corresponds to a vacuum noise.
Definition 1 By canonical commutation relations, we mean that the system variables x(t) satisfy the commutation relations [x i (t), x j (t)] = 2iΘ ij , where Θ is a block diagonal matrix with each diagonal block equal to Such a Θ is said to be canonical.
Without loss of generality, we restrict our attention to quantum plants (1) with canonical commutation relations, this just fixes the choice of basis for x(t). For systems with non-canonical Θ, there exists an equivalent description (1) with canonical Θ which can be obtained by applying the appropriate state transformation (see more details in [2]).
We will consider the class of coherent observers described by QSDEs of the form: (2) (These QSDEs represent an important class of quantum systems considered in [2,3,[13][14][15].) Here ξ(t), dη(t), dv 1 (t) and dv 2 (t) are column vectors of operators with even dimensions n ξ , n η , n v 1 and n v 2 respectively, and n ξ = n η = n x = n y . ξ(t) represents the internal state of the coherent observer which takes the output of the plant dy(t) as its input field(s). dη(t) describes the output of the observer. The coherent observers that we have proposed in above, also have additional inputs driven by quantum vacuum noises. For notational convenience, we separate into dv 1 (t) and dv 2 (t) such that n v 1 = n ξ and n v 2 ≥ 0. These quantum vacuum noises have the following Ito products where F v i is a block diagonal matrix with each block equal to As for the plant, without loss of generality, we restrict our attention to coherent observers with canonical commutation relations. For clarity's sake, from this point onward, we will cease to explicitly include time dependence in our notation. We will use x in place of x(t), etc. however the reader should bear in mind that in general all operators (x, y, dw 1 , etc.) are time dependent. Matrices (A, B 1 , etc.) are time invariant.

Physical Realizability
Not all QSDEs of the form (1) or (2) represent the dynamics of a physically meaningful quantum system. As in [2,15], QSDEs that describe open quantum harmonic oscillators are said to be physically realizable.
We restrict our attention to physically realizable quantum plants and as such the QSDEs (1) are assumed to be physically realizable, however for our results to be meaningful, we will need to ensure that the coherent observers that we propose are also physically realizable.
In some of the literature (e.g. [2]), the term physical realizability is used to describe physically meaningful systems with both classical and quantum degrees of freedom. Here, we are interested in implementing a coherent observer which is a fully quantum system with no classical degrees of freedom.
In the following, we give the definition of physical realizability for QSDEs representing fully quantum systems. Furthermore, we tailor our definition to the class of systems presented by Equation (2) (for a more general definition of physical realizability see [2, Definition 3.1]). Definition 2 The system described by (2) is physically realizable if it has canonical commutation relations.
Moreover, there exists a quadratic Hamiltonian H = 1 2 ξ(0) T Rξ(0), with a real, symmetric, n ξ × n ξ matrix R, and a coupling operator Here, Γ is a (n v 1 + n v 2 + n y ) × (n v 1 + n v 2 + n y ) matrix and . P is the appropriately dimensioned square permutation matrix such that P a 1 a 2 · · · a 2m = a 1 a 3 · · · a 2m−1 a 2 a 4 · · · a 2m and diag(M ) is the appropriately dimensioned square block diagonal matrix with the matrix M occurring along the diagonal. (Note: dimensions of P and diag(M ) can always be determined from the context in which they appear.) Im (.) denotes the imaginary part of a matrix and † denotes the complex conjugate transpose of a matrix.
In order to obtain coherent observers which are physically realizable, we will make use of the following results.
Theorem 1 (See [15,Theorem 3]) Consider an LTI system of the form (2) wherê A,B,Ĉ are given. There exists B v 1 and B v 2 such that the system is physically realizable with canonical commutation matrix Θ, and with n v 2 = r, where r is the rank of the matrix ΘBΘB T Θ − ΘÂ −Â T Θ −Ĉ T ΘĈ . Conversely, suppose that there exists B v 1 and B v 2 such that the system (2) is physically realizable with canonical commutation matrix Θ. Then n v 2 ≥ r, where r is the rank of the matrix This means that it is not possible to choose B v 1 and B v 2 such that the system is physically realizable and the dimension of dv 2 is less than r.
In [15], during the proof of this theorem, we give a method for constructing B v 1 and B v 2 . We described such a method in Appendix A. Again, this method results in the smallest possible dimension for dv 2 (for a givenÂ,B,Ĉ) such that (2) is physically realizable.
Below, we give another theorem that we will need in the following.
Theorem 2 (See [13, Theorem 2]) Consider an LTI system of the form (2), wherê A,B,Ĉ are given and the system commutation matrix Θ is canonical. Suppose that the Riccati equation has a solution X which is skew symmetric and suppose that there exists a real nonsingular matrix T such that X = T T ΘT . Then, there exists a system described by Ã ,B,C with the same transfer function as the system Â ,B,Ĉ which can be physically realized without the B v 2 dv 2 term (i.e. with n v 2 = 0) and where In [14], sufficient conditions are given for the existence of a suitable solution to (3). The accompanying proof leads to a numerical process for obtaining the solution X that we include in Appendix B.

Problem Formulation
The problem we address is to design a coherent observer which we give its formal definition below (see also [18,19]).
Definition 3 A coherent observer is a physically realizable quantum system which has the following property: the mean values of the coherent observer variables track asymptotically the corresponding mean values of the plant.
In this paper, we are designing coherent observers of the form (2) which, when connected to the output of the quantum plant (1), the mean values of the signal component of the output of the coherent observers tracks asymptotically the mean values of the internal states of the quantum plant.

Remark 1
The coherent observer may incorporate additional inputs (other than those connected to the plant outputs) driven by quantum vacuum noises. These may be required to ensure physical realizability.
Finding an optimal estimation of the plant's state is difficult because of the requirement for physical realizability and the constraints that this imposes. We restrict our attention to design of coherent observers which provide suboptimal solutions to such an estimation.

Algorithms to Design Coherent Observers
In this section, we give three algorithms for coherent observer design. The motivation behind these algorithms is to treat the quantum plants classically to obtain Kalman filters and we then obtain physically realizable quantum systems by allowing additional quantum vacuum noises.

Algorithm 1
This is the simplest of the algorithms that we propose. The quantum noises dw 1 , dw 2 driving the plant (1) are treated as classical Wiener processes with intensity  (4) is not physically realizable.
Consider the system: By Theorem 1 there exists B v 1 and B v 2 such that the system (5) is physically realizable. This system is a coherent observer. Furthermore, within the class of quantum systems described by the QSDEs (2), this coherent observer has the minimum number of additional quantum noises (n v 1 + n v 2 ) for our choice of Â ,B,Ĉ . We now describe our algorithm in more detail.
We first obtain a standard Kalman filter (4) (for details of this see for example [16]) Here Q is the solution to the algebraic Riccati equation: Q, if it exists, is the steady state error covariance: describes the intensity of the joint process That is, In particular, Details for constructing B v 1 and B v 2 are included in Appendix A. To see that (5) is a coherent observer for the plant (1), it only remains to be shown that x − ξ converges to zero asymptotically.
The combined plant and observer dynamics are as follows: From (7), making the necessary substitutions, x − ξ satisfies Hence, As a result, x − ξ converges asymptotically to zero if (A − KC) is Hurwitz. The fact that (A − KC) is Hurwitz, follows from the properties of the classical Kalman filter which was used to choose K. Finally, the system (5) so obtained is a coherent observer.
To assess the performance of the coherent observer, we introduce the following performance metric which we wish to minimize: J is obtained by solving the Lyapunov equation where Algorithm 2 This algorithm is a refinement of the first, introducing a free parameter ρ, over which we optimize. The purpose of this parameter is to take into account the impact of the noise terms B v 1 dv 1 (t) and B v 2 dv 2 (t) when designing the Kalman filter. These noise terms are equivalent to additional measurement noise in the plant (1), however they cannot be calculated until after the Kalman filter is designed and hence are not available to the design process. Compared to Algorithm 1, before calculating the Kalman filter, we first introduce an additional term into the plant model (1) to obtain the modified plant Here, dw 3 is a vacuum noise source with Ito product where F w 3 is a block diagonal matrix with each block equal to In effect, we inflate the value of the plant measurement noise when designing the Kalman filter to compensate for the unknown noise terms B v 1 dv 1 (t) and B v 2 dv 2 (t).
We now state Algorithm 2. The following procedure is repeated for different values of ρ > 0: • Obtain the Kalman filter (4) for the modified plant (9) as follows: Here Q is the solution to the Riccati equation (6) with • Obtain B v 1 and B v 2 as in Algorithm 1, such that the system is physically realizable. This system is a coherent observer.
• Calculate the performance metric J as in Algorithm 1 by solving the Lyapunov equation (8). (J is calculated for the actual plant (1) and not for the modified plant (9)).
We then choose the coherent observer (10) that resulted in the least value for J. To see that each iteration results in a coherent observer consider the following: from the properties of the classical Kalman filter, (A − KC) remains Hurwitz for ρ ≥ 0.

Algorithm 3
Our final algorithm attempts to improve performance by reducing the number of additional quantum noises incorporated in the coherent observer. Under certain sufficient conditions it is possible to obtain a coherent observer from a state transformation of the Kalman filter obtained in Algorithm 1. This coherent observer incorporates the minimum number of additional noises possible for a system of the form (2): n v 2 = 0. Algorithm 3 proceeds as follows.
• Attempt to find a transformation T : is physically realizable for someB v 1 . From Theorem 2, if the Riccati equation has a non-singular, real, skew-symmetric solution X, then such a T exists. Sufficient conditions and a construction for T are included in Appendix B. If the sufficient conditions for T are not satisfied, we revert to Algorithm 1.
• The system (11) is a coherent observer.
To see that (11) is a coherent observer it remains to be shown that x −Cξ converges asymptotically to zero which we do now.
The combined plant, observer dynamics can be described as follows Making the appropriate substitutions we obtain From the properties of the Kalman filter (4), (A − KC) is Hurwitz, therefore x −Cξ converges asymptotically to zero for arbitrary initial conditions and (11) is a coherent observer. The performance metric for the coherent observer (11) is the solution to the following Lyapunov equation  Figure 2. Quantum plant and classical observer consisting of heterodyne measurement and a Kalman filter.

Measurement-based (Classical) Observer
In the example which follows we compare the coherent observers designed in the previous section with the following classical observer which consists of heterodyne measurement and a Kalman filter as depicted in figure 2.
The output of the heterodyne measurement is described by the equation where, dw 4 is a vacuum noise source of dimension n w 4 = n w 1 and with Ito product where F w 4 is a block diagonal matrix with each block equal to The following Kalman filter is applied to the output y H of the heterodyne measurement: Here,Â , and Q is the solution to the Riccati equation (6) with By combining equations (1), (12) and (13) we obtain the following dynamics for the combined plant and classical observer: The performance metric for the classical observer is the solution to the following Lyapunov equation: = F w 1,2,4 dt and S w 1,2,4 = Re F w 1,2,4 .

Example
Optical Cavity  Consider the quantum plant depicted in Figure 3. This plant consists of an optical cavity with thermal and vacuum noise inputs. Its dynamics are described by the following QSDEs of the form (1): Here, κ 1 , κ 2 are related to the mirror reflectances. dw 1 is vacuum noise and dw 2 is thermal noise of intensity k n , We consider three scenarios, each with different values for κ 1 , κ 2 . For each scenario, we apply our algorithms to obtain coherent observers across a range of thermal noise intensities k n .   Figure 4 compares the performance metric J for each of our coherent observers with that for a classical observer consisting of heterodyne measurement and a Kalman filter. The classical observer performs best. This is not surprising as each of our coherent observers introduces at least as much additional quantum noise as does the heterodyne measurement in the classical observer. Furthermore, the classical observer is optimal solution with respect to the output of the heterodyne measurement whereas the coherent observers we consider are suboptimal. Notwithstanding this result, it is still of interest to develop tractable methods for designing coherent observers as other considerations may favour the use of coherent observers over measurement-based observers, in particular when the controllers are added.
The performance of Algorithm 1 never exceeds that of Algorithm 2. This is because, in Algorithm 2, ρ = 0 results in the same coherent observer as Algorithm 1. Recall that Algorithm 1 does not take into account the B v 1 and B v 2 terms when designing the Kalman filter. Figure 5 shows the matrix norms for B v 1 and B v 2 for different values of k n . It seems reasonable that as B v 2 becomes more significant, there is greater scope for Algorithm 2 to outperform Algorithm 1. This explains why Algorithm 2 s relative performance increases with k n . Figure 6 shows how the optimal ρ varies with k n in Algorithm 2.
We now turn our attention to Algorithm 3. For small values of k n , a suitable transformation matrix T was found and a coherent observer obtained with n v 2 = 0. In this regime, Algorithm 3 outperforms the other algorithms and produces a coherent observer which approaches the performance of the classical observer. The discontinuity in Algorithm 3 s performance corresponds to the point above which, no suitable T was found. In this regime the algorithm produces the same coherent observer as Algorithm 1. The range of k n for which a suitable T exists is dependent on κ 1 and κ 2 as demonstrated in the scenarios which follow. Note that Algorithm 3 produces a coherent observer with a different value for B v 1 than that from Algorithm 1. In the following scenarios we shall see that in some regimes, despite introducing a smaller number of vacuum noises, Algorithm 3 does not perform better than Algorithm 1.
Finally, we briefly comment on the performance of the observers in the limit as k n approaches zero (that is, as the noise input dw 2 approaches a vacuum noise). For k n = 0, the Kalman filter gain K, obtained in our algorithms, is zero. When dw 2 is a vacuum noise, the output of the plant gives no useful information about the internal state of the plant. In this special case, the optimal coherent observer is the trivial one: a vacuum noise source. See [20] for a discussion of a class of plants driven solely by vacuum noises for which the authors show that the optimal controllers (and by implication the optimal observers) are trivial ones.
7.2. Scenario 2: κ 1 = 0.5; κ 2 = 0.01 Figure 7 shows the performance of the observers obtained for κ 1 = 0.5 and κ 2 = 0.01. (Compared to Scenario 1, mirror 1 is more lossy, while mirror 2 is less lossy.) For these mirrors, Algorithm 3 performs better than Algorithm 1 for greater noise intensities k n . The discontinuity where no suitable state transformation T was found in Algorithm 3 occurs at k n = 69 and is shown in more detail in Figure 8.  Figure 9 shows the performance of the observers obtained for κ 1 = 0.8 and κ 2 = 0.01. Compared to the previous scenarios, mirror 1 is even more lossy. As a result, the discontinuity in Algorithm 3's performance, above which no suitable state transformation T was found, occurs at the increased noise intensity k n = 910. Below this point Algorithm 3 gives a coherent observer with n v 2 = 0 while above this point it gives a coherent observer with n v 2 = 2. Algorithms 1 and 2 give coherent observers with n v 2 = 2 for all considered values of k n .
This scenario demonstrates a region where Algorithm 2 performs better than Algorithm 3 despite the latter giving a coherent observer with less quantum noise sources. This is because, in this region, the impact of the B v 1 term obtained in Algorithm 3 is more significant than the combined impact of both the B v 1 and B v 2 terms in Algorithm 2.
Finally, this scenario suggests that the performance metric J obtained for Algorithms 1 and 2 is not necessarily smooth with respect to k n . Obtaining an explanation for this observation remains the subject of future research.

Conclusions
Like the celebrated Kalman filter in the context of classical feedback control problems, it is envisaged that coherent observers will play a pivotal role in solving coherent quantum feedback control problems. Here, we have proposed three algorithms for the design of coherent observers. The key idea behind each of our algorithms was to first treat the quantum plants classically to obtain a Kalman filter. We then made use of previous results to obtain a physically realizable system by taking the Kalman filter obtained and allowing additional vacuum noise sources in its quantum implementation. Algorithm's 2 and 3 incorporate refinements to Algorithm 1 in an attempt to improve performance.
We compare the performance of the coherent observers obtained with a measurement-based (classical) observer by way of an example involving an optical cavity with thermal and vacuum noise inputs. For each of the scenarios considered, the classical observer performs best. Algorithm 2 always performs at least as well as Algorithm 1. Algorithm 3 can potentially give a coherent observer with a smaller number of quantum vacuum noise inputs than the other algorithms, however this does not guarantee better performance.
physically realizable system. It is not possible to construct B v 1 and B v 2 with smaller n v 2 such that (2) is physically realizable. For further details see [15].
(Here Θ is the canonical commutation matrix of dimension n x × n x ) • Find the rank of the matrixS: n v 2 = rank S .
• Calculate S = i 4S . • Diagonalize S: S = U † DU . Here D is diagonal and U is unitary.
• ConstructD by replacing each element of D with its absolute value.
• Construct B v 1 and B v 2 as follows:

Appendix B.
Here we give a numerical process for obtaining a suitable solution X to the Riccati equation (3) in Theorem 2. The three assumptions in the following, guarantee the existence of the solution X. For further details see [14].
• Find the eigenvalues and eigenvectors of H.
• Assumption 1: That H has no purely imaginary eigenvalues. In practice, this means checking that the real part of each eigenvalue has magnitude greater than some small numerical tolerance.
• Construct the matrix X 1 X 2 such that its columns are the eigenvectors of H that correspond to eigenvalues with negative real part.
• Find the eigenvalues and eigenvectors of X. Hence, construct diagonal Λ with diagonal entries the eigenvalues of X and V with columns the corresponding eigenvectors normalized to length 1.
• Construct the n ξ × n ξ diagonal matrixΛ with alternating diagonal entries i and −i.
• Construct the n ξ × n ξ block diagonal matrixṼ with each diagonal block corresponding to 1 √ 2 1 1 i −i .