Robust Stability of Quantum Systems with a Nonlinear Coupling Operator

# Robust Stability of Quantum Systems with a Nonlinear Coupling Operator

Ian R. Petersen, Valery Ugrinovskii and Matthew R James This work was supported by the Australian Research Council (ARC) and Air Force Office of Scientific Research (AFOSR). This material is based on research sponsored by the Air Force Research Laboratory, under agreement number FA2386-09-1-4089. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the U.S. Government. Ian R. Petersen and Valery Ugrinovskii are with the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra ACT 2600, Australia. {i.r.petersen,v.ugrinovskii}@gmail.comMatthew R. James is with the Research School of Engineering, College of Engineering and Computer Science, The Australian National University, Canberra, ACT 0200, Australia. Email: Matthew.James@anu.edu.au.
###### Abstract

This paper considers the problem of robust stability for a class of uncertain quantum systems subject to unknown perturbations in the system coupling operator. A general stability result is given for a class of perturbations to the system coupling operator. Then, the special case of a nominal linear quantum system is considered with non-linear perturbations to the system coupling operator. In this case, a robust stability condition is given in terms of a scaled strict bounded real condition.

## I Introduction

An important concept in modern control theory is the notion of robust or absolute stability for uncertain nonlinear systems in the form of a Lur’e system with an uncertain nonlinear block which satisfies a sector bound condition; e.g., see [1]. This enables a frequency domain condition for robust stability to be given. This characterization of robust stability enables robust feedback controller synthesis to be carried out using control theory; e.g., see [2]. In a recent paper [3], classical results on robust stability were extended to the case of nonlinear quantum systems with non-quadratic perturbations to the system Hamiltonian. The aim of this paper is to extend classical results on robust stability to the case of nonlinear quantum systems with nonlinear perturbations to the system coupling operator.

In recent years, a number of papers have considered the feedback control of systems whose dynamics are governed by the laws of quantum mechanics rather than classical mechanics; e.g., see [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. In particular, the papers [13, 17] consider a framework of quantum systems defined in terms of a triple where is a scattering matrix, is a vector of coupling operators and is a Hamiltonian operator. The paper [17] then introduces notions of dissipativity and stability for this class of quantum systems. In this paper, we build on the results of [17] to obtain robust stability results for uncertain quantum systems in which the quantum system coupling operator is decomposed as where is a known nominal coupling operator and is a perturbation coupling operator, which is contained in a specified set of coupling operators .

For this general class of uncertain quantum systems, a general stability result is obtained. The paper then considers the case in which the nominal quantum system is a linear quantum system in which the Hamiltonian is a quadratic function of annihilation and creation operators and the coupling operator is a linear function of annihilation and creation operators; e.g., see [7, 8, 10, 11, 16]. In this special case, a robust stability result is obtained in terms of a scaled frequency domain condition.

The remainder of the paper proceeds as follows. In Section II, we define the general class of uncertain quantum systems under consideration. In Section III, we consider a special class of non-linear perturbation coupling operators. In Section IV, we specialize to the case of a linear nominal quantum systems and obtain a robust stability result for this case in which the stability condition is given in terms of a strict bounded real condition dependent on three scaling parameters. In Section V, we present some conclusions.

## Ii Quantum Systems

We consider open quantum systems defined by parameters where ; e.g., see [13, 17]. The corresponding generator for this quantum system is given by

 G(X)=−i[X,H]+LL(X) (1)

where . Here, denotes the commutator between two operators and the notation denotes the adjoint of an operator. Also, is a self-adjoint operator on the underlying Hilbert space referred to as the system Hamiltonian. is the nominal system coupling operator and is referred to as the perturbation coupling operator. Also, is a unitary matrix referred to as the scattering matrix. Throughout this paper, we will assume that . The triple , along with the corresponding generator define the Heisenberg evolution of an operator according to a quantum stochastic differential equation

 dX = (LL(X)−i[X,H])dt +dA∗S†[X,L]+[L∗,X]SdA +tr[(S†XS−X)dΛT];

e.g., see [17]. Here, in the case of operators, the notation denotes the adjoint transpose of a vector or matrix of operators; see also [17] for a definition of the quantities , , and which will not be further considered in this paper. Also, in the case of standard matrices, the notation refers to the complex conjugate transpose of a matrix.

The problem under consideration involves establishing robust stability properties for an uncertain open quantum system for the case in which the perturbation coupling operator is contained in a given set . The main robust stability results presented in this paper will build on the following result from [17].

###### Lemma 1 (See Lemma 3.4 of [17].)

Consider an open quantum system defined by and suppose there exists a non-negative self-adjoint operator on the underlying Hilbert space such that

 G(V)+cV≤λ (2)

where and are real numbers. Then for any plant state, we have

 ⟨V(t)⟩≤e−ct⟨V⟩+λc,  ∀t≥0.

Here denotes the Heisenberg evolution of the operator and denotes quantum expectation; e.g., see [17].

### Ii-a Commutator Decomposition

Given a set of non-negative self-adjoint operators and real parameters , , , , we now define a particular set of perturbation coupling operators . This set is defined in terms of the commutator decomposition

 [V,L2] = w1[V,ζ]−12w2[ζ,[V,ζ]] (3)

for where , and are given scalar operators. We say if the following sector bound condition holds:

 L∗2L2≤1γ2ζ∗ζ+δ1 (4)

and

 w∗1w1≤δ2, (5)
 w∗2w2≤δ3. (6)

Here, we use the convention that for operator inequalities, terms consisting of real constants are interpreted as that constant multiplying the identity operator.

Then, we define

 W1={L2:∃w1, w2, ζ such that (???), (???) and (???)  are satisfied and (???) is satisfied ∀V∈P}. (7)

Using this definition, we obtain the following theorem.

###### Theorem 1

Consider a set of non-negative self-adjoint operators and an open quantum system where and defined in (7). If there exists a and real constants , , such that is a constant and

 −i[V,H]+LL1(V)+(τ212+τ222)L∗1L1 +(δ22τ21+δ22τ24)[V,ζ]∗[V,ζ] +(τ232γ2+τ242γ2+τ252γ2)ζ∗ζ+[V,L1]∗[V,L1]2τ23 +cV≤~λ,

then

 ⟨V(t)⟩ ≤ e−ct⟨V⟩+ ~λ+(δ32τ22+δ32τ25)μ∗μ+(τ232+τ242+τ252)δ1c

for all .

Proof: Let be given and consider defined in (1). Then using (3) and the fact that is self-adjoint,

 G(V) (9) = −i[V,H]+12(L1+L2)∗[V,L1+L2] +12[(L1+L2)∗,V](L1+L2) = −i[V,H] +12(L∗1+L∗2)([V,L1]+w1[V,ζ]+μw2) +12([V,L1]∗+[V,ζ]∗w∗1+μ∗w∗2)(L1+L2) = −i[V,H]+12L∗1[V,L1]+12[V,L1]∗L1 +12L∗1w1[V,ζ]+12[V,ζ]∗w∗1L1 +12μL∗1w2+12μ∗w∗2L1 +12L∗2[V,L1]+12[V,L1]∗L2 +12L∗2w1[V,ζ]+12[V,ζ]∗w∗1L2 +12μL∗2w2+12μ∗w∗2L2.

Now

 0 ≤ (τ1L∗1−[V,ζ]∗w∗1τ1)(τ1L1−w1[V,ζ]τ1) = τ21L∗1L1−L∗1w1[V,ζ]−[V,ζ]∗w∗1L1 +[V,ζ]∗w∗1w1[V,ζ]τ21

and hence

 L∗1w1[V,ζ]+[V,ζ]∗w∗1L1 (10) ≤ τ21L∗1L1+[V,ζ]∗w∗1w1[V,ζ]τ21 ≤ τ21L∗1L1+δ2[V,ζ]∗[V,ζ]τ21

using (5). Also,

 0 ≤ (τ2L∗1−μ∗w∗2τ2)(τ2L1−μw2τ2) = τ22L∗1L1−μL∗1w2−μ∗w∗2L1 +μ∗μw∗2w2τ22

and hence

 μL∗1w2+μ∗w∗2L1 ≤ τ22L∗1L1+μ∗μw∗2w2τ22 (11) ≤ τ22L∗1L1+δ3μ∗μτ22

using (6). Also,

 0 ≤ (τ3L∗2−[V,L1]∗τ3)(τ3L2−[V,L1]τ3) = τ23L∗2L2−L∗2[V,L1]−[V,L1]∗L2 +[V,L1]∗[V,L1]τ23

and hence

 L∗2[V,L1]+[V,L1]∗L2 (12) ≤ τ23L∗2L2+[V,L1]∗[V,L1]τ23 ≤ τ23γ2ζ∗ζ+τ23δ1+[V,L1]∗[V,L1]τ23

using (4). Also,

 0 ≤ (τ4L∗2−[V,ζ]∗w∗1τ4)(τ4L2−w1[V,ζ]τ4) = τ24L∗2L2−L∗2w1[V,ζ]−[V,ζ]∗w∗1L2 +[V,ζ]∗w∗1w1[V,ζ]τ24

and hence

 L∗2w1[V,ζ]+[V,ζ]∗w∗1L2 (13) ≤ τ24L∗2L2+[V,ζ]∗w∗1w1[V,ζ]τ24 ≤ τ24γ2ζ∗ζ+τ24δ1+δ2[V,ζ]∗[V,ζ]τ24

using (4) and (5). Also,

 0 ≤ (τ5L∗2−μ∗w∗2τ5)(τ5L2−μw2τ5) = τ25L∗2L2−μL∗2w2−μ∗w∗2L2 +μ∗μw∗2w2τ25

and hence

 μL∗2w2+μ∗w∗2L2 (14) ≤ τ25L∗2L2+μ∗μw∗2w2τ25 ≤ τ25γ2ζ∗ζ+τ25δ1+δ3μ∗μτ25

using (4) and (6).

Substituting (10), (11), (12), (13) and (14) into (9), it follows that

 G(V) ≤ −i[V,H]+LL1(V)+τ212L∗1L1+δ2[V,ζ]∗[V,ζ]2τ21 (15) +τ222L∗1L1+δ3μ∗μ2τ22 +τ232γ2ζ∗ζ+τ232δ1+[V,L1]∗[V,L1]2τ23 +τ242γ2ζ∗ζ+τ242δ1+δ2[V,ζ]∗[V,ζ]2τ24 +τ252γ2ζ∗ζ+τ252δ1+δ3μ∗μ2τ25.

Then it follows from (1) that

 G(V)+cV≤~λ+(δ32τ22+δ32τ25)μ∗μ+(τ232+τ242+τ252)δ1.

Then the result of the theorem follows from Lemma 1.

## Iii Non-linear Perturbation Coupling Operators

In this section, we define a set of non-linear perturbation coupling operators denoted . For a given set of non-negative self-adjoint operators and real parameters , , , , consider perturbation coupling operators defined in terms of the following power series (which is assumed to converge in some suitable sense)

 L2=f(ζ)=∞∑k=0Skζk=∞∑k=0SkLk. (16)

Here is a scalar operator on the underlying Hilbert space and .

Also, we let

 f′(ζ)=∞∑k=1kSkζk−1, (17)
 f′′(ζ)=∞∑k=1k(k−1)Skζk−2 (18)

and consider the sector bound condition

 f(ζ)∗f(ζ)≤1γ2ζ∗ζ+δ1 (19)

and the conditions

 f′(ζ)∗f′(ζ)≤δ2, (20)
 f′′(ζ)∗f′′(ζ)≤δ3. (21)

Then we define the set as follows:

 W2={L2 of the form (???) such that  conditions (???), (???) and (???) are % satisfied}. (22)

In this section, the set of non-negative self-adjoint operators will be assumed to satisfy the following assumption:

###### Assumption 1

Given any , the quantity

 μ=−12[ζ,[V,ζ]]=−12ζ[V,ζ]+12[V,ζ]ζ

is a constant.

###### Lemma 2

Suppose the set of self-adjoint operators satisfies Assumption 1. Then

 W2⊂W1.

Proof: First, we note that given any and ,

 Vζ = [V,ζ]+ζV; ⋮ Vζk = k∑n=1ζn−1[V,ζ]ζk−n+ζkV. (23)

Also for any such that ,

 [V,ζ]ζ = ζ[V,ζ]+2μ; ⋮ [V,ζ]ζk−n = ζk−n[V,ζ]+2(k−n)ζk−n−1μ. (24)

Therefore using (III) and (III), it follows that

 Vζk = k∑n=1ζn−1ζk−n[V,ζ]+2(k−n)ζn−1ζk−n−1μ +ζkV = k∑n=1ζk−1[V,ζ]+2(k−n)ζk−2μ+ζkV = kζk−1[V,ζ]+k(k−1)ζk−2μ+ζkV,

which holds for any .

Now given any , we have

 [V,Lk] = kζk−1[V,ζ]+k(k−1)ζk−2μ. (25)

Therefore,

 [V,L2] = ∞∑k=0Sk[V,Lk] (26) = f′(ζ)[V,ζ]+f′′(ζ)μ.

Now letting

 w1=f′(ζ), and w2=f′′(ζ), (27)

it follows that condition (3) is satisfied. Furthermore, conditions (4), (5), (6) follow from conditions (19), (20), (21) respectively. Hence, . Since, was arbitrary, we must have .

## Iv The Case of a Linear Nominal System

We now consider the case in which the nominal quantum system corresponds to a linear quantum system; e.g., see [7, 8, 10, 11, 16]. In this case, we assume that is of the form

 H=12[a†aT]M[aa#] (28)

where is a Hermitian matrix of the form

 M=[M1M2M#2M#1]

and , . Here is a vector of annihilation operators on the underlying Hilbert space and is the corresponding vector of creation operators. In the case vectors of operators, the notation refers to the vector of adjoint operators and in the case of complex matrices, this notation refers to the complex conjugate matrix.

The annihilation and creation operators are assumed to satisfy the canonical commutation relations:

 [[aa#],[aa#]†] = [aa#][aa#]† (29) −([aa#]#[aa#]T)T = J

where ; e.g., see [9, 14, 16].

We also assume is of the form

 (30)

where and . Also, we write

 [L1L∗1]=N[aa#]=[N1N2N#2N#1][aa#].

In addition we assume that is of the form

 (31)

where is a positive-definite Hermitian matrix of the form

 P=[P1P2P#2P#1]. (32)

Hence, we consider the set of non-negative self-adjoint operators defined as

 P1={V of the form (???) such that P>0 is a  Hermitian matrix of the form (???)}. (33)

In the linear case, we will consider a specific notion of robust mean square stability.

###### Definition 1

An uncertain open quantum system defined by where with of the form (30), , is any given set, and of the form (28) is said to be robustly mean square stable if there exist constants , and such that for any

 ⟨[a(t)a#(t)]†[a(t)a#(t)]⟩ (34) ≤ c1e−c2t⟨[aa#]†[aa#]⟩+c3  ∀t≥0.

Here denotes the Heisenberg evolution of the vector of operators ; e.g., see [17].

In order to address the issue of robust mean square stability for the uncertain linear quantum systems under consideration, we first require some algebraic identities.

###### Lemma 3

Given , defined as in (28) and defined as in (30), then

 [V,H]= = [aa#]†[PJM−MJP][aa#].

Also,

 LL1(V)= 12L†1[V,L1]+12[L†1,V]L1 = Tr(PJN†[I000]NJ) −12[aa#]†(N†JNJP+PJN†JN)[aa#].

Furthermore,

 [[aa#],[a†aT]P[aa#]]=2JP[aa#].

Proof: The proof of these identities follows via straightforward but tedious calculations using (29).

We now specialize the results of Section II to the case of a linear nominal system with where is defined as in Section III. In this case, we define

 ζ = E1a+E2a# (35) = [E1E2][aa#]=~E[aa#]

where is assumed to be a scalar operator. Then, we show that a sufficient condition for robust mean square stability is the existence of constants , , and such that the following scaled strict bounded real condition is satisfied:

1. The matrix

 F=−iJM−12JN†JN is Hurwitz; (36)
2.  ∥∥¯C(sI−F)−1¯B∥∥∞<1 (37)

where

 ¯C=⎡⎢⎣√τ23+τ24γ~Eτ1~N⎤⎥⎦ (38)

and

 ¯B=[√δ2(1τ21+1τ24)J~E†1τ3J~N†]. (39)

This leads to the following theorem.

###### Theorem 2

Consider an uncertain open quantum system defined by such that where is of the form (30), is of the form (28) and . Furthermore, assume that there exists constants , , and such that the strict bounded real condition (36), (37) is satisfied. Then the uncertain quantum system is robustly mean square stable.

In order to prove this theorem, we require the following lemma.

###### Lemma 4

Given any , then

 μ=−12[ζ,[V,ζ]]=−12~EΣJPJ~ET.

which is a constant. Here, Hence, the set of operators satisfies Assumption 1.

Proof: The proof of this result follows via a straightforward but tedious calculation using (29).

Proof of Theorem 2. If the conditions of the theorem are satisfied, then (37) implies

 ∥∥∥¯C√2(sI−F)−1√2¯B∥∥∥∞<1.

Hence, it follows from the strict bounded real lemma that the matrix inequality

 F†P+PF+2P¯B¯B†P+12¯C†¯C<0.

will have a solution of the form (32); e.g., see [2, 11]. This matrix defines a corresponding operator as in (31). Then using (38) and (39), it follows that we can write.

 F†P+PF +P(2δ2(1τ21+1τ24)J~E†~EJ+2τ23J~N†~NJ)P +τ23+τ242γ2~E†~E+τ212~N†~N<0.

Hence, we can choose and sufficiently small so that

 F†P+PF (40) +P(2δ2(1τ21+1τ24)J~E†~EJ+2τ23J~N†~NJ)P +τ23+τ24+τ252γ2~E†~E+τ21+τ222~N†~N<0.

Now, it follows from (35) that we can write

 ζ∗ζ = (41)

Also, it follows from Lemma 3 that

 [V,ζ]=−2~EJP[aa#].

Hence,

 (42)

Similarly

 (43)

 L∗1L1 = (44)

Hence using Lemma 3, we obtain

 −i[V,H]+LL1(V)+(τ212+τ222)L∗1L1 (45) +(δ22τ21+δ22τ24)[V,ζ]∗[V,ζ] +(τ232γ2+τ242γ2+τ252γ2)ζ∗ζ+[V,L1]∗[V,L1]2τ23 = [aa#]†⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝F†P+PF+2δ2(1τ21+1τ24)PJ~E†~EJP+2τ23PJ~N†~NJP+τ23+τ24+τ252γ2~E†~E+τ21+τ222~N†~N⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠[aa#] +Tr(PJN†[ccI000]NJ)

where .

From this, it follows using (40) that there exists a constant such that condition (1) is satisfied with

 ~λ=Tr(PJN†[[]ccI000]NJ)≥0.

Hence, it follows from Lemma