Quantum Linear Systems Theory

# Quantum Linear Systems Theory

## Abstract

This paper surveys some recent results on the theory of quantum linear systems and presents them within a unified framework. Quantum linear systems are a class of systems whose dynamics, which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs). Such systems commonly arise in the area of quantum optics and related disciplines. Systems whose dynamics can be described or approximated by linear QSDEs include interconnections of optical cavities, beam-spitters, phase-shifters, optical parametric amplifiers, optical squeezers, and cavity quantum electrodynamic systems. With advances in quantum technology, the feedback control of such quantum systems is generating new challenges in the field of control theory. Potential applications of such quantum feedback control systems include quantum computing, quantum error correction, quantum communications, gravity wave detection, metrology, atom lasers, and superconducting quantum circuits.

A recently emerging approach to the feedback control of quantum linear systems involves the use of a controller which itself is a quantum linear system. This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation. This paper discusses recent results concerning the synthesis of H-infinity optimal controllers for linear quantum systems in the coherent control case. An important issue which arises both in the modelling of linear quantum systems and in the synthesis of linear coherent quantum controllers is the issue of physical realizability. This issue relates to the property of whether a given set of QSDEs corresponds to a physical quantum system satisfying the laws of quantum mechanics. The paper will cover recent results relating the question of physical realizability to notions occuring in linear systems theory such as lossless bounded real systems and dual J-J unitary systems.

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## 1 Introduction

Developments in quantum technology and quantum information provide an important motivation for research in the area of quantum feedback control systems; e.g., see [1, 2, 3, 4, 5, 6, 7]. In particular, in recent years, there has been considerable interest in the feedback control and modeling of linear quantum systems; e.g., see [3, 5, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Such linear quantum systems commonly arise in the area of quantum optics; e.g., see [27, 28, 29]. Feedback control of quantum optical systems has applications in areas such as quantum communications, quantum teleportation, and gravity wave detection. In particular, linear quantum optics is one of the possible platforms being investigated for future communication systems (see [30, 31]) and quantum computers (see [32, 33] and [34]). Feedback control of quantum systems aims to achieve closed loop properties such as stability [35, 36], robustness [11, 37], entanglement [18, 38, 39].

Quantum linear system models have been used in the physics and mathematical physics literature since the 1980’s; e.g., see [40, 41, 42, 28, 26]. An important class of linear quantum stochastic models describe the Heisenberg evolution of the (canonical) position and momentum, or annihilation and creation operators of several independent open quantum harmonic oscillators that are coupled to external coherent bosonic fields, such as coherent laser beams; e.g., see [27], [26], [28], [8], [10], [43, 11, 12, 22, 9, 17, 18, 13, 44, 25]). These linear stochastic models describe quantum optical devices such as optical cavities [29], [27], linear quantum amplifiers [28], and finite bandwidth squeezers [28]. Following [11, 12, 22], we will refer to this class of models as linear quantum stochastic systems. In particular, we consider linear quantum stochastic differential equations driven by quantum Wiener processes; see [28]. Further details on quantum stochastic differential equations and quantum Wiener processes can be found in [40, 42, 45].

This paper will survey some of the available results on the feedback control of linear quantum systems and related problems. An important class of quantum feedback control systems involves the use of measurement devices to obtain classical output signals from the quantum system and no quantum measurements is involved. These classical signals are fed into a classical controller which may be implemented via analog or digital electronics and then the resulting control signal act on the quantum system via an actuator. However, some recent papers on the feedback control of linear quantum systems have considered the case in which the feedback controller itself is also a quantum system. Such feedback control is often referred to as coherent quantum control; e.g., see [46, 47, 5, 6, 11, 12, 14, 15, 16, 17, 48]. Due to the limitations imposed by quantum mechanics on the use of quantum measurement, the use of coherent quantum feedback control may lead to improved control system performance. In addition, in many applications, coherent quantum feedback controllers may be preferable to classical feedback controllers due to considerations of speed and ease of implementation.

One motivation for considering such coherent quantum control problems is that coherent controllers have the potential to achieve improved performance since quantum measurements inherently involve the destruction of quantum information; e.g., see [34]. Also, technology is emerging which will enable the implementation of complex coherent quantum controllers (e.g., see [49]) and the coherent controllers proposed in [11] have already been implemented experimentally as described in [17]. Furthermore, coherent controllers implemented using quantum optics have the potential to operate at much higher speeds than classical controllers implemented in analog or digital electronics.

In general, quantum linear stochastic systems represented by linear Quantum Stochastic Differential Equations (QSDEs) with arbitrary constant coefficients need not correspond to physically meaningful systems. In contrast, because classical linear stochastic systems can be implemented at least approximately, using analog or digital electronics, we regard them as always being realizable. Physical quantum systems must satisfy some additional constraints that restrict the allowable values for the system matrices defining the QSDEs. In particular, the laws of quantum mechanics dictate that closed quantum systems evolve unitarily, implying that (in the Heisenberg picture) certain canonical observables satisfy the so-called canonical commutation relations (CCR) at all times. Therefore, to characterize physically meaningful systems, [11] has introduced a formal notion of physically realizable quantum linear stochastic systems and derives a pair of necessary and sufficient characterizations for such systems in terms of constraints on their system matrices.

In the paper [21], the physical realizability results of [14, 15] are extended to the most general class of complex linear QSDEs. It is shown that this class of linear quantum systems corresponds to the class of real linear quantum systems considered in [11] via the use of a suitable state transformation.

The remainder of this paper proceeds as follows. In Section 2, we introduce the class of linear quantum stochastic systems under consideration and consider a number of different representations of these systems. We also introduce a useful special class of linear quantum systems which was considered in [14, 15, 16]. In Section 3, we consider the issue of physical realizability for the class of linear quantum systems under consideration. In Section 4, we will consider the problem of coherent quantum controller synthesis. In Section 5, we present some conclusions.

## 2 Linear Quantum System Models

In this section, we formulate the class of linear quantum system models under consideration. These linear quantum system models take the form of quantum stochastic differential equations which are derived from the quantum harmonic oscillator.

### 2.1 Quantum Harmonic Oscillators

We begin by considering a collection of independent quantum harmonic oscillators which are defined on a Hilbert space ; e.g., see [50, 42, 25]. Elements of the Hilbert space , are the standard complex valued wave functions arising in quantum mechanics where is a spatial variable. Corresponding to this collection of harmonic oscillators is a vector of annihilation operators

 a=⎡⎢ ⎢ ⎢ ⎢⎣a1a2⋮an⎤⎥ ⎥ ⎥ ⎥⎦. (1)

Each annihilation operator is an unbounded linear operator defined on a suitable domain in by

 (aiψ)(x)=1√2xiψ(x)+1√2∂ψ(x)∂xi

where is contained in the domain of the operator . The adjoint of the operator is denoted and is referred to as a creation operator. The operators and are such that the following cannonical commutation relations are satisfied

 [ai,a∗j]=aia∗j−a∗jai=δij (2)

where denotes the kronecker delta multiplied by the identity operator on the Hilbert space . We also have the commutation relations

 [ai,aj]=0, [a∗i,a∗j]=0. (3)

For a general vector of operators

 g=⎡⎢ ⎢ ⎢ ⎢ ⎢⎣g1g2⋮gn⎤⎥ ⎥ ⎥ ⎥ ⎥⎦,

on , we use the notation

 g#=⎡⎢ ⎢ ⎢ ⎢ ⎢⎣g∗1g∗2⋮g∗n⎤⎥ ⎥ ⎥ ⎥ ⎥⎦,

to denote the corresponding vector of adjoint operators. Also, denotes the corresponding row vector of operators , and . Using this notation, the canonical commutation relations (2), (3) can be written as

 [[aa#],[aa#]†] = [aa#][aa#]† (4) −([aa#]#[aa#]T)T = [I00−I].

A state on our system of quantum harmonic oscillators is defined by a density operator which is a self-adjoint positive-semidefinite operator on with ; e.g., see [34]. Corresponding to a state and an operator on is the quantum expectation

 ⟨g⟩=tr(ρg).

A state on the system is said to be Gaussian with positive-semidefinite covariance matrix and mean vector if given any vector ,

 ⟨exp(i[u†uT][aa#])⟩ = Unknown environment '%

e.g., see [25, 50]. Here, denotes the complex conjugate of the complex vector , denotes the transpose of the complex vector , and denotes the complex conjugate transpose of the complex vector .

Note that in the zero mean case, , the covariance matrix satisfies

 Q=⟨[aa#][aa#]†⟩.

In the special case in which the covariance matrix is of the form

 Q=[I000]

and the mean , the system is said to be in the vacuum state. In the sequel, it will be assumed that the state on the system of harmonic oscillators is a Gaussian vacuum state. The state on the system of harmonic oscillators plays a similar role to the probability distribution of the initial conditions of a classical stochastic system.

The quantum harmonic oscillators described above are assumed to be coupled to external independent quantum fields modelled by bosonic annihilation field operators which are defined on separate Fock spaces defined over for each field operator [40, 42, 45, 39]. For each annihilation field operator , there is a corresponding creation field operator , which is defined on the same Fock space and is the operator adjoint of . The field operators are adapted quantum stochastic processes with forward differentials

 dAj(t)=Aj(t+dt)−Aj(t)

and

 dA∗j(t)=A∗j(t+dt)−A∗j(t)

that have the quantum Itô products [40, 42, 45, 39]:

 dAj(t)dAk(t)∗ = δjkdt; dA∗j(t)dAk(t) = 0; dAj(t)dAk(t) = 0; dA∗j(t)dA∗k(t) = 0.

The field annilation operators are also collected into a vector of operators defined as follows:

 A(t)=⎡⎢ ⎢ ⎢ ⎢ ⎢⎣A1(t)A2(t)⋮Am(t)⎤⎥ ⎥ ⎥ ⎥ ⎥⎦.

For each , the corresponding system state on the Fock space is assumed to be a Gaussian vacuum state which means that given any complex valued function , then

 ⟨exp(i∫∞0ui(t)∗dAi(t)+i∫∞0ui(t)dAi(t)∗)⟩ = exp(−12∫∞0|u(t)|2dt);

e.g., see [40, 42, 45, 25].

In order to describe the joint evolution of the quantum harmonic oscillators and quantum fields, we first specify the Hamiltonian operator for the quantum system which is a Hermitian operator on of the form

 H=12[a†aT]M[aa#]

where is a Hermitian matrix of the form

 M=[M1M2M#2M#1]

and , . Here, denotes the complex conjugate transpose of the complex matrix , denotes the transpose of the complex matrix , and denotes the complex conjugate of the complex matrix . Also, we specify the coupling operator for the quantum system to be an operator of the form

 L=[N1N2][aa#]

where and . Also, we write

 [LL#]=N[aa#]=[N1N2N#2N#1][aa#].

In addition, we define a scattering matrix which is a unitary matrix . These quantities then define the joint evolution of the quantum harmonic oscillators and the quantum fields according to a unitary adapted process (which is an operator valued function of time) satisfying the Hudson-Parthasarathy QSDE [40, 42, 45, 23]:

 dU(t) = (tr[(S−I)TdΛ(t)]+dA(t)†L−L†SdA(t) −(iH+12L†L)dt)U(t); U(0)=I,

where . Here, the processes for are adapted quantum stochastic processes referred to as gauge processes, and the forward differentials have the quantum Itô products:

 dΛjk(t)dΛj′k′(t) = δkj′dΛjk′(t); dAj(t)dΛkl(t) = δjkdAl(t); dΛjkdAl(t)∗ = δkldA∗j(t).

Then, using the Heisenberg picture of quantum mechanics, the harmonic oscillator operators evolve with time unitarily according to

 ai(t)=U(t)∗aiU(t)

for . Also, the linear quantum system output fields are given by

 Aouti(t)=U(t)∗Ai(t)U(t)

for .

We now use the fact that for any adapted processes and satisfying a quantum Itô stochastic differential equation, we have the quantum Itô rule

 dX(t)Y(t)=X(t)dY(t)+dX(t)Y(t)+dX(t)dY(t);

e.g., see [42]. Using the quantum Itô rule and the quantum Itô products given above, as well as exploiting the canonical commutation relations between the operators in , the following QSDEs decribing the linear quantum system can be obtained (e.g., see [25]):

 da(t) = dU(t)∗aU(t) = [F1F2][a(t)a(t)#]dt +[G1G2][dA(t)dA(t)#]; a(0) = a; dAout(t) = dU(t)∗A(t)U(t) (5) = [H1H2][a(t)a(t)#]dt +[K1K2][dA(t)dA(t)#],

where

 F1 = −iM1−12(N†1N1−NT2N#2); F2 = −iM2−12(N†1N2−NT2N#1); G1 = −N†1S; G2 = NT2S#; H1 = N1; H2 = N2; K1 = S; K2 = 0. (6)

From this, we can write

 [da(t)da(t)#] = F[a(t)a(t)#]dt+G[dA(t)dA(t)#]; [dAout(t)dAout(t)#] = H[a(t)a(t)#]dt+K[dA(t)dA(t)#],

where

 F = [F1F2F#2F#1];  G=[G1G2G#2G#1]; H = [H1H2H#2H#1];  K=[K1K2K#2K#1]. (8)

Also, the equations (2.1) can be re-written as

 F = −iJM−12JN†JN; G = −JN†[S00−S#]; H = N; K = [S00S#]; (9)

where

 J=[I00−I].

Note that matrices of the form (2.1) occur commonly in the theory of linear quantum systems. It is straightforward to establish the following lemma which characterizes matrices of this form.

###### Lemma 1

A matrix satisfies

 [R1R2R3R4]=[R1R2R#2R#1]

if and only if

 RΣ=ΣR#

where

 Σ=[0II0].

We now consider the case when the initial condition in the QSDE (2.1) is no longer the vector of annihilation operators (1) but rather a vector of linear combinations of annihilation operators and creation operators defined by

 ~a=T1a+T2a#

where

 T=[T1T2T#2T#1]∈C2n×2n

is non-singular. Then, it follows from (4) that

 [[~a~a#],[~a~a#]†] = [~a~a#][~a~a#]†−([~a~a#]#[~a~a#]T)T = T[~a~a#][~a~a#]†T† −⎛⎝T#([aa#]#[aa#]T)TTT⎞⎠T = T⎛⎜ ⎜ ⎜ ⎜ ⎜⎝[aa#][aa#]†−([aa#]#[aa#]T)T⎞⎟ ⎟ ⎟ ⎟ ⎟⎠T† = Θ

where

 Θ = TJT† (10) = [T1T†1−T2T†2T1TT2−T2TT1T#2T†1−T#1T†2T#2TT2−T#1TT1].

The relationship

 [[~a~a#],[~a~a#]†]=Θ (11)

is referred to as a generalized commutation relation [14, 15, 16]. Also, the covariance matrix corresponding to is given by

 ~Q = ⟨[~a~a#][~a~a#]†⟩ = T[I000]T† = [T1T†1T1TT2T#2T†1T#2TT2].

In terms of the variables , the QSDEs, (2.1) can be rewritten as

 [d~a(t)d~a(t)#] = ~F[~a(t)~a(t)#]dt+~G[dA(t)dA(t)#]; [dAout(t)dAout(t)#] = ~H[~a(t)~a(t)#]dt+~K[dA(t)dA(t)#],

where

 ~F = [~F1~F2~F#2~F#1]=TFT−1; ~G = [~G1~G2~G#2~G#1]=TG; ~H = [~H1~H2~H#2~H#1]=HT−1; ~K = [~K1~K2~K#2~K#1]=K. (13)

Now, we can re-write the operators and defining the above collection of quantum harmonic oscillators in terms of the variables as

 Unknown environment '%

where

 ~M=(T†)−1MT−1,  ~N=NT−1. (14)

Here,

 ~M=[~M1~M2~M#2~M#1],  ~N=[~N1~N2~N#2~N#1]. (15)

Furthermore, equations (2.1), (2.1) and (14) can be combined to obtain

 ~F = −iΘ~M−12Θ~N†J~N; ~G = −Θ~N†[S00−S#]; ~H = ~N; ~K = [S00S#]. (16)

Note that since is unitary, it follows that

 ~KJ~K† = [S00S#][I00−I][S†00ST] (17) = [S00S#][S†00−ST] = [SS†00−S#ST]=J.

Also,

 ~K~K† = [S00S#][S†00ST] (18) = [SS†00S#ST]=I.

Indeed these two properties characterize all matrices non-singular satisfying which are of the form given in (2.1). Let the nonsingular matrix be such that , , and . It follows from Lemma 1 that we can write

 K=[K1K2K#2K#1].

Also, implies

 KK† = [K1K2K#2K#1][K†1KT2K†2KT1] = [K1K†1+K2K†2K1KT2+K2KT1K#2K†1+K#1K†2K#2KT2+K#1KT1] = [I00I]

and implies

 KJK† = [K1K2K#2K#1][I00−I][K†1KT2K†2KT1] = [K1K2K#2K#1][K†1KT2−K†2−KT1] = [K1K†1−K2K†2K1KT2−K2KT1K#2K†1−K#1K†2K#2KT2−K#1KT1] = [I00−I].

The block of these two equations imply and . Hence . Therefore, and . From this, it follows that the matrix must be of the form given in (2.1).

The QSDEs (2.1), (2.1), (2.1) define the general class of linear quantum systems considered in this paper. Such quantum systems can be used to model a large range of devices and networks of devices arising in the area of quantum optics including optical cavities, squeezers, optical parametric amplifiers, cavity QED systems, beam splitters, and phase shifters; e.g., see [3, 5, 6, 11, 17, 19, 22, 24, 26, 27, 28, 29, 48].

### 2.2 Annihilation operator linear quantum systems

An important special case of the linear quantum systems (2.1), (2.1), (2.1) corresponds to the case in which the Hamiltonian operator and coupling operator depend only of the vector of annihilation operators and not on the vector of creation operators . This class of linear quantum systems is considered in [14, 20, 15, 16, 19, 51, 17] and can be used to model “passive” quantum optical devices such as optical cavities, beam splitters, phase shifters and interferometers.

This class of linear quantum systems corresponds to the case in which , , and . In this case, the linear quantum system can be modelled by the QSDEs

 d~a(t) = ~F~a(t)dt+~GdA(t) dAout(t) = ~H~a(t)dt+~KdA(t) (19)

where

 ~F = −iΘ1~M1−12Θ1~N†1~N1; ~G = −Θ1~N†1S; ~H = ~N1; ~K = S; Θ1 = T1T†1>0. (20)

### 2.3 Position and momentum operator linear quantum systems

Note that the matrices in the general QSDEs (2.1), (2.1) are in general complex. However, it is possible to apply a particular change of variables to the system (2.1) so that all of the matrices in the resulting transformed QSDEs are real. This change of variables is defined as follows:

 [qp] = Φ[aa#]; [Q(t)P(t)] = Φ[A(t)A(t)#]; [Qout(t)Pout(t)] = Φ[Aout(t)Aout(t)#] (21)

where the matrices have the form

 Φ=[II−iIiI] (22)

and have the appropriate dimensions. Here is a vector of the self-adjoint position operators for the system of harmonic oscillators and is a vector of momentum operators; e.g., see [21, 11, 12, 39]. Also, and are the vectors of position and momentum operators for the quantum noise fields acting on the system of harmonic oscillators. Furthermore, and are the vectors of position and momentum operators for the output quantum noise fields.

It follows from (22) that

 ΦΦ†=2I (23)

and hence

 Φ−1Φ−†=12I. (24)

Rather than applying the transformations (2.3) to the quantum linear system (2.1) which satisfies the canonical commutation relations (4), corresponding transformations can be applied to the quantum linear system (2.1) which satisfies the generalized commutation relations (11). These transformations are as follows:

 [~q~p] = Φ[~a~a#]; [Q(t)P(t)] = Φ[A(t)A(t)#]; [Qout(t)Pout(t)] = Φ[Aout(t)Aout(t)#]. (25)

When these transformations are applied to the quantum linear system (2.1), this leads to the following real quantum linear system:

 [d~q(t)d~p(t)] = A[~p(t)~q(t)]dt+B[dQ(t)dP(t)]; [dQout(t)dPout(t)] = C[~q(t)~p(t)]dt+D[dP(t)dQ(t)],

where

 A = Φ~FΦ−1 = 12⎡⎢⎣~F1+~F#1+~F2+~F#2−i(~F1−~F#1)−i(~F2−~F#2) i(~F1−~F#1)−i(~F2−~F#2)~F1+~F#1−~F2−~F#2⎤⎥⎦; B = Φ~GΦ−1 = 12⎡⎢⎣~G1+~G#1+~G2+~G#2−i(~G1−~G#1)−i(~G2−~G#2) i(~G1−~G#1)−i(~G2−~G#2)~G1+~G#1−