开放系统动力学的密度张量体系：恢复噪声
A density tensor hierarchy for open system dynamics: retrieving the noise
摘要：
我们介绍了一个开放系统动力学的密度张量层次，恢复信息波动损失通过减少密度矩阵。对于由经典概率分布引起的涨落的情况，该层次由纯态密度矩阵元素的乘积的期望形成，并且可以由简单的生成函数来简洁地概括。对于量子系统与处于整体纯态的量子环境相互作用时产生量子涨落的情形，相应的层次定义为全密度矩阵的系统矩阵元的乘积的环境迹。只有量子噪声层级中最低的成员才能在实验上直接测量。纯态密度矩阵的单位迹和幂等性质意味着张量层次的下降关系，即在适当的张量指数对的收缩下，$n$张量的顺序与$n-1$张量的顺序相关。作为说明经典概率分布形式的例子，我们考虑了一个量子系统演化的伊藤随机和跳跃过程薛定谔方程。作为例子，我们考虑了无限质量布朗粒子的碰撞布朗运动和弱耦合Born-Markov主方程，说明了量子涨落情形下相应的迹形式.在不同的专门化下，后者给出了量子光学主方程和Caldeira-Leggett主方程的推广层次。作为密度张量的进一步应用，我们对比了减少和不减少状态矢量的随机薛定谔方程，并讨论了为什么耦合到量子环境的量子系统表现得像后者。
英文摘要：
We introduce a density tensor hierarchy for open system dynamics, that recovers information about fluctuations lost in passing to the reduced density matrix. For the case of fluctuations arising from a classical probability distribution, the hierarchy is formed from expectations of products of pure state density matrix elements, and can be compactly summarized by a simple generating function. For the case of quantum fluctuations arising when a quantum system interacts with a quantum environment in an overall pure state, the corresponding hierarchy is defined as the environmental trace of products of system matrix elements of the full density matrix. Only the lowest member of the quantum noise hierarchy is directly experimentally measurable. The unit trace and idempotence properties of the pure state density matrix imply descent relations for the tensor hierarchies, that relate the order $n$ tensor, under contraction of appropriate pairs of tensor indices, to the order $n-1$ tensor. As examples to illustrate the classical probability distribution formalism, we consider a quantum system evolving by Itô stochastic and by jump process Schrödinger equations. As examples to illustrate the corresponding trace formalism in the quantum fluctuation case, we consider collisional Brownian motion of an infinite mass Brownian particle, and the weak coupling Born-Markov master equation. In different specializations, the latter gives the hierarchies generalizing the quantum optical master equation and the Caldeira--Leggett master equation. As a further application of the density tensor, we contrast stochastic Schrödinger equations that reduce and that do not reduce the state vector, and discuss why a quantum system coupled to a quantum environment behaves like the latter.
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