A general framework for complete positivity
A general framework for complete positivity
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DOI：
10.1007/s11128-015-1148-0
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发表时间：
2016-01-01
影响因子：
2.5
通讯作者：
Lidar, Daniel A.
中科院分区：
文献类型：
--
作者：
Dominy, Jason M.;Shabani, Alireza;Lidar, Daniel A.
量子动力学的完全正性常常被看作是物理性的试金石；然而，众所周知，相关的初始状态不一定会产生完全正向的演化。这一观察结果在过去二十年中激发了许多研究，试图确定完全积极的必要和充分条件。在这里，我们描述了一个完整和一致的数学框架，用于讨论和分析开放量子系统相关初始态的完全正性。这种形式主义建立在几个简单的公理之上，并且足够普遍，可以包含Pechakas （Phys Rev Lett 73: 1060-1062, 1994）的所有先前的方法。关键的观察结果是，系统上具有相同化简状态的初始系统浴状态必须在所有允许的幺正算子下演化到系统上具有相同化简状态的系统浴状态，以确保系统上的诱导动态映射是定义良好的。一旦这个一致性条件被施加，相关的概念，如赋值映射和动态映射是唯一定义的。通常，动态映射不能应用于任意的系统状态，而只能应用于适当定义的物理域中的状态。我们证明了问题的约束性质产生了不是一种而是三种不相等的完全正性类型。利用这个框架，我们阐明了最近使用量子不和谐和量子数据处理不等式为完全正性提供条件的尝试的局限性。特别是，我们纠正了我们中的两个人（Shabani和Lidar在《物理学评论》102:100402-100404,2009）提出的主张，即不和谐的消失是完全正性所必需的，并解释说它只对一类特定的初始状态有效。这个问题仍然是开放的，可能需要新的视角和新的数学工具。这里提出的形式主义可能是朝那个方向迈出的一步。
Complete positivity of quantum dynamics is often viewed as a litmus test for physicality; yet, it is well known that correlated initial states need not give rise to completely positive evolutions. This observation spurred numerous investigations over the past two decades attempting to identify necessary and sufficient conditions for complete positivity. Here, we describe a complete and consistent mathematical framework for the discussion and analysis of complete positivity for correlated initial states of open quantum systems. This formalism is built upon a few simple axioms and is sufficiently general to contain all prior methodologies going back to Pechakas (Phys Rev Lett 73: 1060-1062, 1994). The key observation is that initial system-bath states with the same reduced state on the system must evolve under all admissible unitary operators to system-bath states with the same reduced state on the system, in order to ensure that the induced dynamical maps on the system are well defined. Once this consistency condition is imposed, related concepts such as the assignment map and the dynamical maps are uniquely defined. In general, the dynamical maps may not be applied to arbitrary system states, but only to those in an appropriately defined physical domain. We show that the constrained nature of the problem gives rise to not one but three inequivalent types of complete positivity. Using this framework, we elucidate the limitations of recent attempts to provide conditions for complete positivity using quantum discord and the quantum data processing inequality. In particular, we correct the claim made by two of us (Shabani and Lidar in Phys Rev Lett 102: 100402-100404, 2009) that vanishing discord is necessary for complete positivity, and explain that it is valid only for a particular class of initial states. The problem remains open, and may require fresh perspectives and new mathematical tools. The formalism presented herein may be one step in that direction.