A Popov Stability Condition for Uncertain Linear Quantum Systems
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 timevarying uncertainties and nonlinearities whereas the Popov criterion only applies to static timeinvariant 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 selfadjoint operator on the underlying Hilbert space referred to as the nominal Hamiltonian and is a selfadjoint 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.
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 nonnegative selfadjoint 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 nonnegative selfadjoint 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.
Iia Commutator Decomposition
We now consider a set of selfadjoint perturbation Hamiltonians . For a given set of nonnegative selfadjoint operators , a set of Popov scaling parameters , a selfadjoint 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 nonnegative selfadjoint 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 (IIA), we have
(11)  
Now
since is selfadjoint. This confirms that the operator on the right hand side of the above identity is a selfadjoint operator. Therefore, the following inequality follows from the second equation (IIA):
(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 selfadjoint 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 (IIA) 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)  
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 positivedefinite Hermitian matrix of the form
(26) 
Hence, we consider the set of nonnegative selfadjoint 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
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.
Proof: The proof of these identities follows via straightforward but tedious calculations using (23).
Lemma 5
Proof: The proof of this result follows via straightforward but tedious calculations using (23).
Lemma 6
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:

The matrix defined in (30) is Hurwitz.

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.
V Illustrative Example
In this section, we consider an example of an open quantum system with