Multi-point Gaussian States, Quadratic–Exponential Cost Functionals, and Large Deviations Estimates for Linear Quantum Stochastic Systems

Abstract

This paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost functional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. Such functionals are related to more conservative behaviour and robustness of systems with respect to statistical uncertainty, which makes the challenging problems of their computation and minimization practically important. To this end, we obtain an integro-differential equation for the time evolution of the quadratic–exponential functional, which is different from the original quantum risk-sensitive performance criterion employed previously for measurement-based quantum control and filtering problems. Using multi-point Gaussian quantum states for the past history of the system variables and their first four moments, we discuss a quartic approximation of the cost functional and its infinite-horizon asymptotic behaviour. The computation of the asymptotic growth rate of this approximation is reduced to solving two algebraic Lyapunov equations. Further approximations of the cost functional, based on higher-order cumulants and their growth rates, are applied to large deviations estimates in the form of upper bounds for tail distributions. We discuss an auxiliary classical Gaussian–Markov diffusion process in a complex Euclidean space which reproduces the quantum system variables at the level of covariances but has different fourth-order cumulants, thus showing that the risk-sensitive criteria are not reducible to quadratic–exponential moments of classical Gaussian processes. The results of the paper are illustrated by a numerical example and may find applications to coherent quantum risk-sensitive control problems, where the plant and controller form a fully quantum closed-loop system, and other settings with nonquadratic cost functionals.

Appendices

Appendix A: Commutator of Quadratic Polynomials of Quantum Variables Satisfying CCRs

Since the commutator $\mathrm {ad}_{\xi }$, associated with a given operator $\xi$, is a derivation, then repeated application of this property leads to

$$\begin{aligned} {[}ab,c d]&= [ab,c]d+ c[ab,d] \nonumber \\&= ([a,c]b + a[b,c])d + c([a,d]b + a[b,d])\nonumber \\&= [a,c]b d + a[b ,c]d +c[a,d]b + ca[b ,d] \end{aligned}$$

(A1)

for any operators a, b, c, d. Hence, if these operators satisfy CCRs (that is, their pairwise commutators are identity operators up to scalar factors), then the right-hand side of (A1) is a quadratic polynomial of a, b, c, d. Therefore, quadratic polynomials of such operators form a Lie algebra with respect to the commutator. For the purposes of Sect. 4, we will provide a version of (A1) for quadratic polynomials of operators satisfying the CCRs.

Lemma 4

Suppose $a:=(a_j)$, $b:=(b_k)$, $c:=(c_{\ell })$, $d:=(d_r)$ are vectors of self-adjoint operators which satisfy the CCRs

$$\begin{aligned} \left[ \begin{bmatrix} a\\ b \end{bmatrix}, \begin{bmatrix} c^{\mathrm{T}}&d^{\mathrm{T}} \end{bmatrix} \right] := \begin{bmatrix} [a,c^{\mathrm{T}}]&\quad [a,d^{\mathrm{T}}]\\ [b,c^{\mathrm{T}}]&\quad [b,d^{\mathrm{T}}] \end{bmatrix} = 2i \begin{bmatrix} \varTheta _{11}&\quad \varTheta _{12} \\ \varTheta _{21}&\quad \varTheta _{22} \end{bmatrix}, \end{aligned}$$

(A2)

where $\varTheta _{jk}$ are real matrices. Also, let $F:= (f_{jk})$ and $G:=(g_{\ell r})$ be appropriately dimensioned complex matrices which specify the bilinear forms

$$\begin{aligned} a^{\mathrm{T}}F b =\sum _{j,k}f_{jk}a_jb_k, \qquad c^{\mathrm{T}} G d = \sum _{\ell ,r}g_{\ell r}c_{\ell }d_r. \end{aligned}$$

(A3)

Then their commutator is also a quadratic polynomial of the operators which is computed as

$$\begin{aligned}{}[a^{\mathrm{T}}F b, c^{\mathrm{T}}G d] = 2i \begin{bmatrix} a^{\mathrm{T}}&\quad b^{\mathrm{T}}&\quad c^{\mathrm{T}} \end{bmatrix} \begin{bmatrix} 0&\quad 0&\quad F \varTheta _{21} G \\ 0&\quad 0&\quad F^{\mathrm{T}} \varTheta _{11} G \\ G \varTheta _{22}^{\mathrm{T}} F^{\mathrm{T}}&\quad G \varTheta _{12}^{\mathrm{T}} F&\quad 0 \end{bmatrix} \begin{bmatrix} a \\ b \\ d \end{bmatrix}.\qquad \end{aligned}$$

(A4)

In the case $a=b$ (when the vectors a and b are identical), (A4) takes the form

$$\begin{aligned}{}[a^{\mathrm{T}}F a,c^{\mathrm{T}}G d] = 4i \begin{bmatrix} a^{\mathrm{T}}&c^{\mathrm{T}} \end{bmatrix} \begin{bmatrix} 0&\quad {\mathbf {S}}(F)\varTheta _{11} G\\ G \varTheta _{12}^{\mathrm{T}} {\mathbf {S}}(F)&\quad 0 \end{bmatrix} \begin{bmatrix} a \\ d \end{bmatrix}, \end{aligned}$$

(A5)

where ${\mathbf {S}}(F):= \frac{1}{2}(F+F^{\mathrm{T}})$ denotes the symmetrizer of square matrices. $\square$

Proof

By recalling the bilinearity of the commutator, applying (A1) to the operators $a_j$, $b_k$, $c_{\ell }$, $d_r$ in (A3) and using the CCRs (A2), it follows that

$$\begin{aligned} {[}a^{\mathrm{T}}F b,c^{\mathrm{T}}G d]&= \sum _{j,k,\ell ,r} f_{jk}g_{\ell r} [a_jb_k, c_{\ell }d_r] \nonumber \\&= \sum _{j,k,\ell ,r} f_{jk}g_{\ell r} \big ( [a_j,c_{\ell }]b_k d_r + a_j[b_k ,c_{\ell }]d_r+c_{\ell }[a_j,d_r]b_k + c_{\ell }a_j[b_k ,d_r] \big ) \nonumber \\&= 2i \big ( b^{\mathrm{T}} F^{\mathrm{T}} \varTheta _{11} Gd + a^{\mathrm{T}} F \varTheta _{21} Gd + c^{\mathrm{T}}G \varTheta _{12}^{\mathrm{T}} Fb + c^{\mathrm{T}}G \varTheta _{22}^{\mathrm{T}} F^{\mathrm{T}}a \big ). \end{aligned}$$

(A6)

The right-hand side of (A6) is a quadratic function of the operators whose vector-matrix form is given by (A4). If $a=b$, then F in (A3) is a square matrix, and the matrices in (A2) satisfy $\varTheta _{1k}=\varTheta _{2k}$ for every $k=1,2$. In this case, (A6) reduces to

$$\begin{aligned}{}[a^{\mathrm{T}}F a,c^{\mathrm{T}}G d]&= 2i \big ( a^{\mathrm{T}} (F^{\mathrm{T}} \varTheta _{11}+F \varTheta _{21}) Gd+ c^{\mathrm{T}}G (\varTheta _{12}^{\mathrm{T}} F+\varTheta _{22}^{\mathrm{T}} F^{\mathrm{T}})a \big )\\&= 2i \big ( a^{\mathrm{T}} (F^{\mathrm{T}} +F )\varTheta _{11} Gd + c^{\mathrm{T}}G \varTheta _{12}^{\mathrm{T}}( F+F^{\mathrm{T}})a \big ) \\&= 4i \big ( a^{\mathrm{T}} {\mathbf {S}}(F)\varTheta _{11} Gd + c^{\mathrm{T}}G \varTheta _{12}^{\mathrm{T}}{\mathbf {S}}(F)a \big ), \end{aligned}$$

which establishes (A5). $\square$

Similar commutation relations hold for bilinear forms of annihilation and creation operators (see, for example, [29, Appendix B] and [61, Lemma 4.2]) and are used in the context of Schwinger’s theorems on exponentials of such forms [10].

Appendix B: Covariance of Quadratic Functions of Gaussian Quantum Variables

For the purposes of Sect. 5, we will need the following lemma on the covariance of bilinear forms of Gaussian quantum variables.

Lemma 5

Suppose the self-adjoint quantum variables, constituting the vectors a, b, c, d in Lemma 4, are in a zero-mean Gaussian state. Then the covariance of the bilinear forms in (A3), specified by complex matrices F and G, can be computed as

$$\begin{aligned} \mathbf {cov}( a^{\mathrm{T}}F b, c^{\mathrm{T}}G d ) = \left\langle \overline{F\mathbf {cov}(b,d)}, \mathbf {cov}(a,c)G \right\rangle + \left\langle \overline{F\mathbf {cov}(b,c)}, \mathbf {cov}(a,d)G^{\mathrm{T}} \right\rangle .\nonumber \\ \end{aligned}$$

(B1)

In the case when $a=b$ and $c=d$, the relation (B1) reduces to

$$\begin{aligned} \mathbf {cov}( a^{\mathrm{T}}F a, c^{\mathrm{T}}G c ) = 2 \left\langle \overline{{\mathbf {S}}(F)}, \mathbf {cov}(a,c) {\mathbf {S}}(G) \mathbf {cov}(a,c)^{\mathrm{T}} \right\rangle , \end{aligned}$$

(B2)

where ${\mathbf {S}}$ is the symmetrizer. $\square$

Proof

The mean value of the first bilinear form in (A3) is computed as

$$\begin{aligned} {\mathbf {E}}(a^{\mathrm{T}}F b) = \sum _{j,k} f_{jk} {\mathbf {E}}(a_jb_k) = \mathrm {Tr}( F^{\mathrm{T}} {\mathbf {E}}(ab^{\mathrm{T}}) ) = \left\langle {\overline{F}}, \mathbf {cov}(a,b) \right\rangle , \end{aligned}$$

(B3)

where ${\mathbf {E}}(ab^{\mathrm{T}}) = \mathbf {cov}(a,b)$ since the underlying quantum variables are assumed to have zero mean values. By a similar reasoning, the second bilinear form in (A3) has the mean value

$$\begin{aligned} {\mathbf {E}}(c^{\mathrm{T}}G d) = \left\langle {\overline{G}}, \mathbf {cov}(c,d) \right\rangle . \end{aligned}$$

(B4)

Application of the Wick–Isserlis theorem [35, Theorem 1.28 on pp. 11–12] (see also [28] and [44, p. 122]) to the fourth-order mixed moment of the zero-mean Gaussian quantum variables $a_j$, $b_k$, $c_{\ell }$, $d_r$ leads to

$$\begin{aligned} {\mathbf {E}}( a^{\mathrm{T}}F b c^{\mathrm{T}}G d )&= \sum _{j,k,\ell ,r} f_{jk}g_{\ell r} {\mathbf {E}}(a_j b_k c_{\ell } d_r) \nonumber \\&= \sum _{j,k,\ell ,r} f_{jk}g_{\ell r} \big ( \mathbf {cov}(a_j,b_k) \mathbf {cov}(c_{\ell },d_r) + \mathbf {cov}(a_j,c_{\ell }) \mathbf {cov}(b_k,d_r) \nonumber \\&\quad + \mathbf {cov}(a_j,d_r) \mathbf {cov}(b_k,c_{\ell }) \big ) \nonumber \\&= \mathrm {Tr}(F^{\mathrm{T}}\mathbf {cov}(a,b)) \mathrm {Tr}(G^{\mathrm{T}}\mathbf {cov}(c,d)) + \mathrm {Tr}((F\mathbf {cov}(b,d))^{\mathrm{T}} \mathbf {cov}(a,c)G) \nonumber \\&\quad + \mathrm {Tr}((F\mathbf {cov}(b,c))^{\mathrm{T}} \mathbf {cov}(a,d)G^{\mathrm{T}}) \nonumber \\&= \left\langle {\overline{F}}, \mathbf {cov}(a,b) \right\rangle \left\langle {\overline{G}}, \mathbf {cov}(c,d) \right\rangle + \left\langle \overline{F\mathbf {cov}(b,d)}, \mathbf {cov}(a,c)G \right\rangle \nonumber \\&\quad + \left\langle \overline{F\mathbf {cov}(b,c)}, \mathbf {cov}(a,d)G^{\mathrm{T}} \right\rangle . \end{aligned}$$

(B5)

The covariance of the bilinear forms (A3) can now be computed by combining (B3)–(B5) as

$$\begin{aligned} \mathbf {cov}( a^{\mathrm{T}}F b, c^{\mathrm{T}}G d )&= {\mathbf {E}}( a^{\mathrm{T}}F b c^{\mathrm{T}}G d ) - {\mathbf {E}}( a^{\mathrm{T}}F b ) {\mathbf {E}}( c^{\mathrm{T}}G d )\\&= \left\langle \overline{F\mathbf {cov}(b,d)}, \mathbf {cov}(a,c)G \right\rangle + \left\langle \overline{F\mathbf {cov}(b,c)}, \mathbf {cov}(a,d)G^{\mathrm{T}} \right\rangle , \end{aligned}$$

which establishes (B1). Application of (B1) to the particular case $a=b$ and $c=d$ leads to

$$\begin{aligned} \mathbf {cov}( a^{\mathrm{T}}F a, c^{\mathrm{T}}G c )&= \left\langle \overline{F\mathbf {cov}(a,c)}, \mathbf {cov}(a,c)G \right\rangle + \left\langle \overline{F\mathbf {cov}(a,c)}, \mathbf {cov}(a,c)G^{\mathrm{T}} \right\rangle \nonumber \\&= 2 \left\langle \overline{F\mathbf {cov}(a,c)}, \mathbf {cov}(a,c){\mathbf {S}}(G) \right\rangle \nonumber \\&= 2 \left\langle {\overline{F}}, \mathbf {cov}(a,c) {\mathbf {S}}(G) \mathbf {cov}(a,c)^{\mathrm{T}} \right\rangle \nonumber \\&= 2 \left\langle \overline{{\mathbf {S}}(F)}, \mathbf {cov}(a,c) {\mathbf {S}}(G) \mathbf {cov}(a,c)^{\mathrm{T}} \right\rangle , \end{aligned}$$

(B6)

thus proving (B2). In (B6) use has also been made of the symmetry of the matrix $\mathbf {cov}(a,c) {\mathbf {S}}(G) \mathbf {cov}(a,c)^{\mathrm{T}}$, the orthogonality of the subspaces of symmetric and antisymmetric matrices, and the fact that the symmetrizer commutes with the complex conjugation: ${\mathbf {S}}({\overline{F}} ) = \overline{{\mathbf {S}}(F)}$. $\square$

Appendix C: An Averaging Lemma with Multifactor Convolutions

For the purposes of the cumulant growth rate results of Theorem 4 in Sect. 6, we provide an averaging lemma which involves multifactor convolutions (145).

Lemma 6

Suppose $f_1, \ldots , f_r$ are appropriately dimensioned complex matrix-valued functions on the real line which are bounded and absolutely integrable, with $r\geqslant 2$. Then

$$\begin{aligned} \lim _{t\rightarrow +\infty } \Big ( \frac{1}{t} \int _{[0,t]^r} \mathop {\overrightarrow{\prod }}_{k=1}^r f_k(t_k-t_{k+1}) \mathrm {d}t_1 \times \cdots \times \mathrm {d}t_r \Big )&= (f_1* \cdots *f_r) (0) \nonumber \\&= \frac{1}{2\pi } \int _{-\infty }^{+\infty } \mathop {\overrightarrow{\prod }}_{k=1}^r F_k(\lambda ) \mathrm {d}\lambda , \end{aligned}$$

(C1)

where $t_{r+1} := t_1$, and $F_k(\lambda ):= \int _{-\infty }^{+\infty } f_k(t)\mathrm {e}^{-i\lambda t} \mathrm {d}t$ denotes the Fourier transform of $f_k$. $\square$

Proof

Since the integrand on the left-hand side of (C1) depends on the integration variables $t_1, \ldots , t_r$ only through their differences, they can be translated so as to represent the integral in the form

$$\begin{aligned} \int _{[0,t]^r} \mathop {\overrightarrow{\prod }}_{k=1}^r f_k(t_k-t_{k+1}) \mathrm {d}t_1 \times \cdots \times \mathrm {d}t_r = \int _0^t g_t(t_1) \mathrm {d}t_1, \end{aligned}$$

(C2)

where

$$\begin{aligned} g_t(t_1)&:= \int _{[0,t]^{r-1}} \mathop {\overrightarrow{\prod }}_{k=1}^r f_k(t_k-t_{k+1}) \mathrm {d}t_2 \times \cdots \times \mathrm {d}t_r \nonumber \\&= \int _{[-t_1, t-t_1]^{r-1}} \mathop {\overrightarrow{\prod }}_{k=1}^r f_k(\tau _k-\tau _{k+1}) \mathrm {d}\tau _2 \times \cdots \times \mathrm {d}\tau _r. \end{aligned}$$

(C3)

Here, use is made of the new integration variables $\tau _k := t_k-t_1$ for all $k=2, \ldots , r$ together with the corresponding convention $\tau _1 = \tau _{r+1}=0$. The right-hand side of (C3) is organized as the convolution $(f_1* \cdots *f_r)(0)$ (evaluated at 0) except that the integration is restricted to the cube $[-t_1, t-t_1]^{r-1}$. Now, the fulfillment of the inclusions $t_1 \in [0, t]$ and $(\tau _2, \ldots , \tau _r) \in [-t_1, t-t_1]^{r-1}$ is equivalent to $t_1$ belonging to the (possibly empty) interval

$$\begin{aligned} - \min (0, \tau _2, \ldots , \tau _r) \leqslant t_1 \leqslant t - \max (0, \tau _2, \ldots , \tau _r) \end{aligned}$$

(C4)

of length

$$\begin{aligned} \max \big ( 0,\, t - \max (0, \tau _2, \ldots , \tau _r) + \min (0, \tau _2, \ldots , \tau _r) \big ) = t h_t(\tau _2, \ldots , \tau _r). \end{aligned}$$

(C5)

Here,

$$\begin{aligned} h_t(\tau _2, \ldots , \tau _r) := \chi _t \big ( \max (0, \tau _2, \ldots , \tau _r) - \min (0, \tau _2, \ldots , \tau _r) \big ) \end{aligned}$$

(C6)

inherits from the function $\chi _t$ in (105) the properties of being bounded by and convergent to 1 as $t\rightarrow +\infty$ for any given $\tau _2, \ldots , \tau _r \in {\mathbb {R}}$. By combining (C2)–(C6), it follows that

$$\begin{aligned}&\frac{1}{t} \int _{[0,t]^r} \mathop {\overrightarrow{\prod }}_{k=1}^r f_k(t_k-t_{k+1}) \mathrm {d}t_1 \times \cdots \times \mathrm {d}t_r \nonumber \\&\quad = \int _{{\mathbb {R}}^{r-1}} h_t(\tau _2, \ldots , \tau _r) \mathop {\overrightarrow{\prod }}_{k=1}^r f_k(\tau _k-\tau _{k+1}) \mathrm {d}\tau _2 \times \cdots \times \mathrm {d}\tau _r \nonumber \\&\quad \rightarrow \int _{{\mathbb {R}}^{r-1}} \mathop {\overrightarrow{\prod }}_{k=1}^r f_k(\tau _k-\tau _{k+1}) \mathrm {d}\tau _2 \times \cdots \times \mathrm {d}\tau _r, \qquad \mathrm{as}\ t\rightarrow +\infty , \end{aligned}$$

(C7)

where Lebesgue’s dominated convergence theorem is applicable since the functions $f_1, \ldots , f_r$ are bounded and absolutely integrable (and hence, so are their convolutions). The limit in (C7) is $(f_1* \cdots *f_r)(0)$, which establishes the first of the equalities (C1), with the second of them following from the convolution theorem. $\square$