高斯量子过程的二次指数泛函
Quadratic-exponential functionals of Gaussian quantum processes
摘要：
本文研究具有位置-动量正则对易关系的量子过程的二次积分函数的指数矩。这样的二次指数泛函（QEF）出现在玻色子场驱动的开放量子谐振子（OQHO）的控制问题的鲁棒性能标准。我们开发了一个随机表示的QEF使用Karhunen-Loeve展开的量子过程的有界时间间隔上的特征基的两点换向器内核，与非交换的位置动量对系数。这种表示法与特定的量子态无关，并且采用辅助经典高斯随机过程的平均，该过程的协方差算子由换向器核指定。这使得QEF与量子过程的矩生成泛函相关，并为多点高斯态计算。对于平稳高斯量子过程，我们建立了一个频域公式的QEF率的量子协方差核的傅里叶变换与三角函数的组合。为了进行近似和数值计算，得到了定量教育基金比率关于风险敏感性参数的微分方程。QEF也适用于大的偏差和最坏情况下的均方成本界限的OQHO在存在统计不确定性与量子相对熵的描述。
英文摘要：
This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen-Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.
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