A Popov Stability Condition for Uncertain Linear Quantum Systems

A Popov Stability Condition for Uncertain Linear Quantum Systems

Matthew R James, Ian R. Petersen and Valery Ugrinovskii This work was supported by the Australian Research Council (ARC) under projects DP110102322, DP1094650 and FL110100020, and the 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. Matthew R. James is with the ARC Centre for Quantum Computation and Communication Technology, Research School of Engineering, College of Engineering and Computer Science, The Australian National University, Canberra, ACT 0200, Australia. Email: Matthew.James@anu.edu.au.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.com. The work of Valery Ugrinovskii was carried out in part while he was a visitor at the Australian National University.
Abstract

This paper considers a Popov type approach to the problem of robust stability for a class of uncertain linear quantum systems subject to unknown perturbations in the system Hamiltonian. A general stability result is given for a general class of perturbations to the system Hamiltonian. Then, the special case of a nominal linear quantum system is considered with quadratic perturbations to the system Hamiltonian. In this case, a robust stability condition is given in terms of a frequency domain condition which is of the same form as the standard Popov stability condition.

I Introduction

This paper builds on the previous papers [1, 2, 3] which consider the problem of robust stability analysis for open quantum systems subject to perturbations in either the system Hamiltonian or coupling operator, which together define the dynamics of the quantum system. The results of these papers can be regarded as extensions of the classical small gain theorem for robust stability to the case of quantum systems. The main contribution of this paper is a result which can be regarded as an extension of the classical Popov criterion for absolute stability to the case of open quantum systems. In particular, we extend the result of [2], in which the perturbations to the system Hamiltonian are uncertain quadratic perturbations, to obtain a corresponding Popov robust stability result.

The small gain theorem and the Popov criterion for absolute stability are two of the most useful tests for robust stability and nonlinear system stability; e.g., see [4]. Both of these stability tests consider a Lur’e system which is the feedback interconnection between a linear time invariant system and a sector bounded nonlinearity or uncertainty. The key distinction between the small gain theorem and the Popov criterion is that the small gain theorem establishes absolute stability via the use of a fixed quadratic Lyapunov function whereas the Popov criterion relies on a Lyapunov function of the Lur’e Postnikov form which involves the sum of a quadratic term and a term dependent on the integral of the nonlinearity itself. The small gain theorem can be used to establish stability in the presence of time-varying uncertainties and nonlinearities whereas the Popov criterion only applies to static time-invariant nonlinearities. However, the Popov criterion is less conservative than the small gain theorem. Hence, we are motivated to obtain a quantum Popov stability criterion in order to obtain less conservative results.

The study of quantum feedback control theory has been the subject of increasing interest in recent years; e.g., see [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In particular, the papers [14, 18] 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 [18] then introduces notions of dissipativity and stability for this class of quantum systems. As in the papers [1, 3], the results of this paper build on the stability results of [18] to obtain robust stability results for uncertain quantum systems in which the quantum system Hamiltonian is decomposed as where is a known nominal Hamiltonian and is a perturbation Hamiltonian, which is contained in a specified set of Hamiltonians .

For this general class of uncertain quantum systems, the paper first obtains a general abstract version of the Popov stability criterion which requires finding a Lyapunov type operator to satisfy an operator inequality. The paper then considers the case in which the nominal Hamiltonian is a quadratic function of annihilation and creation operators and the coupling operator vector is a linear function of annihilation and creation operators. This case corresponds to a nominal linear quantum system; e.g., see [8, 9, 11, 12, 17]. Also, it is assumed that the perturbation Hamiltonian is quadratic but uncertain. In this special case, a robust stability stability criterion is obtained in terms of a frequency domain condition which takes the same form as the classical Popov stability criterion.

The remainder of the paper proceeds as follows. In Section II, we define the general class of uncertain quantum systems under consideration. In this section, we also present a general Popov type stability result for this class of quantum systems. In Section III, we consider a class of uncertain quadratic perturbation Hamiltonians. In Section IV, we specialize to the case of linear nominal quantum systems and obtain a robust stability result for this case in which the stability condition is a frequency domain condition in the same form as the classical Popov stability condition. In Section V we present an illustrative example involving a quantum system arising from an optical parametric amplifier. In Section VI, we present some conclusions.

Ii Quantum Systems

We consider open quantum systems defined by parameters where ; e.g., see [14, 18]. The corresponding generator for this quantum system is given by

(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 nominal Hamiltonian and is a self-adjoint operator on the underlying Hilbert space referred to as the perturbation Hamiltonian. The triple , along with the corresponding generators define the Heisenberg evolution of an operator according to a quantum stochastic differential equation; e.g., see [18].

The problem under consideration involves establishing robust stability properties for an uncertain open quantum system for the case in which the perturbation Hamiltonian is contained in a given set . Using the notation of [18], the set defines a set of exosystems. This situation is illustrated in the block diagram shown in Figure 1.

Fig. 1: Block diagram representation of an open quantum system interacting with an exosystem.

The main robust stability results presented in this paper will build on the following result from [18].

Lemma 1 (See Lemma 3.4 of [18].)

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

(2)

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

Here denotes the Heisenberg evolution of the operator and denotes quantum expectation; e.g., see [18].

We will also use following result, which is a slight modification of Theorem 3.1 and Lemma 3.4 of [18].

Lemma 2

Consider an open quantum system defined by and suppose there exists non-negative self-adjoint operators and on the underlying Hilbert space such that the following quantum dissipation inequality holds

(3)

where is a real number. Then for any plant state, we have

(4)

Here denotes the Heisenberg evolution of the operator and denotes quantum expectation; e.g., see [18].

Proof The proof is similar to the proof of Lemma 3.4 in [18]. In a similar manner we obtain from (3)

(5)

where denotes vacuum expectation operator. Taking the vacuum expectation on both sides, and noting that results in

for any in the underlying Hilbert space. Then for any corresponding plant state (i.e., for such that ) we have

Then (4) follows in a standard manner.

Ii-a Commutator Decomposition

We now consider a set of self-adjoint perturbation Hamiltonians . For a given set of non-negative self-adjoint operators , a set of Popov scaling parameters , a self-adjoint operator , which is the nominal Hamiltonian, a coupling operator , and for a real parameter , this set is defined in terms of the commutator decompositions

for all and , where and are given operator vectors of the same dimension. Here, the notation denotes the adjoint transpose of a vector of operators. In addition, the notation denotes the vector of adjoint operators for a given vector of operators.

Then, the set will be defined in terms of the sector bound condition

(7)

where is a given constant. That is, we define

(8)

Using this definition, we obtain the following theorem.

Theorem 1

Consider a set of non-negative self-adjoint operators , an open quantum system and an observable where and defined in (8). Suppose there exists a and real constants , such that

(9)

Then

Here denotes the Heisenberg evolution of the operator .

Proof: Let and be given such that the conditions of the theorem are satisfied and consider defined in (1). Then

(10)

Using the decomposition in the first equation (II-A), we have

(11)

Now

since is self-adjoint. This confirms that the operator on the right hand side of the above identity is a self-adjoint operator. Therefore, the following inequality follows from the second equation (II-A):

(12)

Also, note that

(13)

Furthermore,

This implies that

Using this inequality and (12), we have

(14)

Then it follows from (7) and (9) that

The result of the theorem then follows from Lemma 2.

Iii Quadratic Perturbations of the Hamiltonian

We consider a set of quadratic perturbation Hamiltonians of the form

(15)

where is a Hermitian matrix of the form

(16)

, and is a given vector of operators. Here, in the case of complex matrices, the notation refers to the complex conjugate transpose of a matrix. Also, in the case of complex matrices, the notation refers to the complex conjugate matrix. In addition, for this case we assume that .

The matrix is subject to the bounds

(17)

Then we define

(18)

Using this definition, we obtain the following lemma.

Lemma 3

Suppose that is a constant vector. Then, for any set of self-adjoint operators ,

Proof: Given any , let and Hence, Then, for any and , let and we have

since is a scalar operator and is a constant matrix. Also,

Hence,

Similarly,

(19)

In addition,

(20)

and similarly

Now using (19), (20), (III) and the assumption that is a constant vector, it follows that

It then follows from (17) that

Therefore we can conclude that both of the conditions in (II-A) are satisfied with . Also, condition (17) implies

and

which implies (7). Hence, . Therefore, .

Iv The Linear Case

We now consider the case in which the nominal quantum system corresponds to a linear quantum system; e.g., see [8, 9, 11, 12, 17]. In this case, we assume that is of the form

(22)

where is a Hermitian matrix of the form

and , . Here is a vector of annihilation operators on the underlying Hilbert space and is the corresponding vector of creation operators. The annihilation and creation operators are assumed to satisfy the canonical commutation relations:

(23)

where ; e.g., see [10, 15, 17].

In addition, we assume is of the form

(24)

where and . Also, we write

In addition, we assume that is of the form

(25)

where is a positive-definite Hermitian matrix of the form

(26)

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

(27)

In the linear case, we also let and hence we can write

(28)

We will also consider a specific notion of robust mean square stability.

Definition 1

An uncertain open quantum system defined by where with of the form (22), , and of the form (24) is said to be robustly mean square stable if for any , there exist constants , and such that

(29)

Here denotes the Heisenberg evolution of the vector of operators ; e.g., see [18].

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 4

Given , defined as in (22) and defined as in (24), then

Also,

Proof: The proof of these identities follows via straightforward but tedious calculations using (23).

Lemma 5

With the variable defined as in (28) and defined as in (24), then

which is a constant vector. Here,

Proof: The proof of this result follows via straightforward but tedious calculations using (23).

Lemma 6

With the variable defined as in (28), defined in (22) and defined as in (24), then

where

(30)

Furthermore,

Proof: The proofs of these equations follows via straightforward but tedious calculations using (23).

We will show that a sufficient condition for robust mean square stability when is the existence of a constant , such that the following conditions are satisfied:

  1. The matrix defined in (30) is Hurwitz.

  2. The transfer function

    (31)

    satisfies the strict positive real (SPR) condition

    (32)

    for all .

This leads to the following theorem.

Theorem 2

Consider an uncertain open quantum system defined by such that where is of the form (22), is of the form (24) and . Furthermore, assume that there exist a constant such that the matrix defined in (30) is Hurwitz and the frequency domain condition (32) is satisfied. Then the uncertain quantum system is robustly mean square stable.

Proof of Theorem 2. If the conditions of the theorem are satisfied, then the transfer function is strictly positive real. However, this transfer function has a state space realization

where is defined as in (30),

(33)

and

(34)

It now follows using the strict positive real lemma that the linear matrix inequality

(35)

will have a solution of the form (26); e.g., see [4]. This matrix defines a corresponding operator as in (25). Furthermore, it is straightforward to verify that . Hence, using Schur complements, it follows from (35) that

Now using Lemma 6 we have

Hence using Lemma 4, we obtain

(37)

where

is defined in (30), is defined in (33) and is defined in (34). From this, it follows using Lemma 5, Lemma 3, (IV), and a similar argument to the proof of Theorem 1 that

where

It follows from (IV) that . Hence using (17), it follows that there exists a constant such that the condition

is satisfied. Therefore, it follows from Lemma 1, Lemma 3, (17) and that

(38)

Hence, the condition (29) is satisfied with , and .

Observation 1

A useful special case of the above result occurs when the QSDEs describing the nominal open quantum linear system depend only on annihilation operators and not on the creation operators; e.g., see [11, 12]. This case corresponds to the case of and . Also, we assume that . In this case, we calculate the matrix in (30) to be

where . Also, we calculate the transfer function matrix in (31) to be

where .

We now consider the case in which is a SISO transfer function. In this case, the condition that the matrix in (30) is Hurwitz reduces to the condition that the matrix

(39)

is Hurwitz. Also, the the SPR condition (32) reduces to the following conditions:

(40)
(41)

for all . The conditions (40), (41) can be tested graphically producing a plot of versus with as a parameter. Such a parametric plot is referred to as the Popov plot; e.g., see [4]. Then, the conditions (40), (41) will be satisfied if and only if the Popov plot lies between two straight lines of slope and with -axis intercepts ; see Figure 2.

\psfrag

Im \psfragRe \psfragslopeslope \psfragg4 \psfragmg4 \psfragarallowable region

Fig. 2: Allowable region for the Popov plot.

V Illustrative Example

In this section, we consider an example of an open quantum system with