A Popov Stability Condition for Uncertain Linear Quantum Systems

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 $(S,L,H)$ where $S$ is a scattering matrix, $L$ is a vector of coupling operators and $H$ 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 $H=H_1+H_2$ where $H_1$ is a known nominal Hamiltonian and $H_2$ is a perturbation Hamiltonian, which is contained in a specified set of Hamiltonians $W$.

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 $H_1$ 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.

We consider open quantum systems defined by parameters $(S,L,H)$ where $H=H_1+H_2$; e.g., see [14, 18]. The corresponding generator for this quantum system is given by

where $L\left(X\right)=\frac{1}{2}{L}^{\ast }[X,L]+\frac{1}{2}[L^{\ast },X]L$. Here, $[X,H]=XH-HX$ denotes the commutator between two operators and the notation ${}^{\ast }$ denotes the adjoint of an operator. Also, $H_1$ is a self-adjoint operator on the underlying Hilbert space referred to as the nominal Hamiltonian and $H_2$ is a self-adjoint operator on the underlying Hilbert space referred to as the perturbation Hamiltonian. The triple $(S,L,H)$, along with the corresponding generators define the Heisenberg evolution $X\left(t\right)$ of an operator $X$ 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 $W_1$. Using the notation of [18], the set $W_1$ 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].

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

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

where $E_t$ denotes vacuum expectation operator. Taking the vacuum expectation $E_0$ on both sides, and noting that $E_0{E}_t=E_0$ results in

for any $\psi$ in the underlying Hilbert space. Then for any corresponding plant state (i.e., for $\psi$ such that $⟨\psi ,\psi ⟩=1$) we have

Then (4) follows in a standard manner. $\square$

We consider a set $W_2$ of quadratic perturbation Hamiltonians of the form

where $\Delta \in C^{2m\times 2m}$ is a Hermitian matrix of the form

${\Delta }_1={\Delta }_1^{†}$, ${\Delta }_2={\Delta }_2^T$ and $\zeta$ 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 $\Theta =[0,\infty )$.

The matrix $\Delta$ is subject to the bounds

Then we define

Using this definition, we obtain the following lemma.

In addition,

and similarly

Now using (19), (20), (III) and the assumption that $[z,L]$ 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 $\beta =\frac{4}{\gamma }$. Also, condition (17) implies

and

which implies (7). Hence, $H_2\in W_1$. Therefore, $W_2\subset W_1$. $\square$

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 $H_1$ is of the form

where $M\in C^{2n\times 2n}$ is a Hermitian matrix of the form

and $M_1=M_1^{†}$, $M_2=M_2^T$. Here $a$ is a vector of annihilation operators on the underlying Hilbert space and $a^{\#}$ is the corresponding vector of creation operators. The annihilation and creation operators are assumed to satisfy the canonical commutation relations:

where $J=\left[\begin{array}{cc}I& 0\\ 0& -I\end{array}\right]$; e.g., see [10, 15, 17].

In addition, we assume $L$ is of the form

where $N_1\in C^{1\times n}$ and $N_2\in C^{1\times n}$. Also, we write

In addition, we assume that $V$ is of the form

where $P\in C^{2n\times 2n}$ is a positive-definite Hermitian matrix of the form

Hence, we consider the set of non-negative self-adjoint operators $P_1$ defined as

In the linear case, we also let $\zeta =E_{1}a+E_2{a}^{\#}$ and hence we can write

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

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.