论文标题
连续时间随机步行量子测量和量子随机滤波和控制的分数方程的建模
Continuous time random walks modeling of quantum measurement and fractional equations of quantum stochastic filtering and control
论文作者
论文摘要
最初在量子随机演算的框架中开发的量子随机滤波的主要方程在得出后，作为在适当缩放下离散测量的Markov模型的限制。在现代物理学的许多分支中，将随机步行建模扩展到连续的随机步行（CTRW）建模，在离散事件之间的时间被视为非指数。在本文中，我们将CTRW建模应用于连续的量子测量结果，从而产生了量子滤波的新分数进化方程，从而获得了开放系统的量子力学的新分数方程。相关的量子控制问题和游戏证明了Riemannian歧管上的分数汉密尔顿 - 雅各比 - 贝尔曼（HJB）方程式描述。与现有文本相比，我们通过修改方式提供了标准量子过滤方程的全面推导，（i）提供了明确的收敛速率（通过以前开发的martingales方法不可用），并且（ii）允许CTRWS基本结果的直接应用CTRWS来推断最终的分量过滤式化度。
Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.
