对于每个不可因子化的系统-环境幺正演化，线性约化动力学的偏离总是会发生
Deviation from linear reduced dynamics always occurs for each non-factorisable system-environment unitary evolution
中文总结 AI 辅助
该研究表明，对于不可因子化的系统-环境幺正演化，无法通过任意初始态实现线性约化动力学，需选初始态子集，且正但非完全正的约化动力学意味着另一演化的约化动力学非线性。
中文摘要 AI 辅助
在最简单的近似中，与环境E相互作用的量子系统S的约化动力学被认为由完全正映射给出，但一般而言并非如此。实际上，系统的约化动力学通常甚至不是线性的。约化动力学是否线性由两个因素决定：系统-环境的可能初始态集合$\boldsymbol{\rho}_{SE}$，以及联合系统-环境幺正演化U。当U可因子化为$U=U_S\boldsymbol{\bigotimes}U_E$时，我们可选择$\boldsymbol{\rho}_{SE}$为所有系统-环境密度算子的集合D；换句话说，当U可因子化时，对于系统-环境的任意初始态$\boldsymbol{\rho}_{SE}$，系统S的约化动力学是线性的（实际上是幺正的）。我们证明该结果不能推广到任何不可因子化的U：对于整个系统-环境的任何不可因子化幺正演化U，必须将$\boldsymbol{\rho}_{SE}$选为D的真子集才能实现线性约化动力学。作为附带结果，考虑满足$\text{Tr}_E \boldsymbol{\rho}_{SE}=\text{Tr}_E D$的系统-环境可能初始态凸集，我们证明：当某一系统-环境幺正演化$U_1$对应的系统约化动力学是正的但非完全正的，这意味着另一$U_2$对应的约化动力学不是线性的。
英文摘要
In the simplest approximation, the reduced dynamics of a quantum system $S$ interacting with its environment $E$ is considered to be given by a completely positive map. But, in general, this is not the case. In fact, the reduced dynamics of the system in not even linear, in general. Whether the reduced dynamics is linear or not is determined by two factors: the set of possible initial states of the system-environment $\mathcal{S}=\left\lbrace ρ_{SE} \right\rbrace $, and the joint system-environment unitary evolution $U$. When $U$ is factorisable as $U=U_S\otimes U_E$, then we can choose $\mathcal{S}=\mathcal{D}$, where $\mathcal{D}$ is the set of all system-environment density operators. In other words, when $U$ is factorisable, the reduced dynamics of the system $S$ is linear (in fact unitary) for arbitrary initial state of the system-environment $ρ_{SE}$. We show that this result cannot be generalized to any non-factorisable $U$: For any non-factorisable unitary evolution of the whole system-environment $U$, the set $\mathcal{S}$ must be chosen as a proper subset of $\mathcal{D}$ to achieve linear reduced dynamics. As a byproduct, considering a convex set of possible initial states of the system-environment $\mathcal{S}$ such that $\mathrm{Tr}_{E} \ \mathcal{S}=\mathrm{Tr}_{E} \ \mathcal{D}$, we show that when the reduced dynamics of the system, for one system-environment unitary evolution $U_1$, is positive, but not completely positive, this implies that reduced dynamics is not linear for another $U_2$.
