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Parametric randomization, complex symplectic factorizations, and quadratic-exponential functionals for Gaussian quantum states

机译：参数随机化，复杂的辛分解，高斯量子状态的二次指数函数

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摘要

This paper combines probabilistic and algebraic techniques for computing quantum expectations of operator exponentials (and their products) of quadratic forms of quantum variables in Gaussian states. Such quadratic-exponential functionals (QEFs) resemble quantum statistical mechanical partition functions with quadratic Hamiltonians and are also used as performance criteria in quantum risk-sensitive filtering and control problems for linear quantum stochastic systems. We employ a Lie-algebraic correspondence between complex symplectic matrices and quadratic-exponential functions of system variables of a quantum harmonic oscillator. The complex symplectic factorizations are used together with a parametric randomization of the quasi-characteristic or moment-generating functions according to an auxiliary classical Gaussian distribution. This reduces the QEF to an exponential moment of a quadratic form of classical Gaussian random variables with a complex symmetric matrix and is applicable to recursive computation of such moments.
机译：本文结合了高斯态在高斯态中量子变量二次形式的算子指数（及其产品）计算量子预期的概率和代数技术。这种二次指数函数（QEFs）类似于二次哈密顿人的量子统计机械分区功能，并且还用作线性量子随机系统量子风险敏感滤波和控制问题的性能标准。我们在复杂的辛矩阵和量子谐波振荡器的系统变量的二次指数函数之间采用了谎言对应关系。根据辅助经典高斯分布，复杂的辛分解与准特征或力矩产生功能的参数随机化一起使用。这将QEF减少到具有复杂对称矩阵的古典高斯随机变量的二次形式的指数时刻，并且适用于这种时刻的递归计算。

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