近似量子马尔可夫链
Approximate quantum Markov chains
摘要：
这本书是量子马尔可夫链的介绍，并解释了这个概念是如何连接到一个丢失的量子力学系统如何从相关的子系统中恢复的问题。为了实现这一目标，我们加强了数据处理不等式，使其揭示了一个关于重建丢失信息的陈述。理解量子马尔可夫链行为的主要困难来自于量子力学算符一般不对易的事实。因此，我们首先解释了两种技术如何处理非交换矩阵：谱捏方法和复杂的插值理论。一旦读者熟悉了这些技术，一个新的不等式提出了著名的黄金-汤普森不等式扩展到任意多个矩阵。这个不等式是理解近似量子马尔可夫链的关键因素，它回答了自1973年以来一直开放的矩阵分析问题，即，如果Lieb的三重矩阵不等式可以推广到三个以上的矩阵。最后，我们仔细讨论了近似量子马尔可夫链的性质及其含义。
英文摘要：
This book is an introduction to quantum Markov chains and explains how this concept is connected to the question of how well a lost quantum mechanical system can be recovered from a correlated subsystem. To achieve this goal, we strengthen the data-processing inequality such that it reveals a statement about the reconstruction of lost information. The main difficulty in order to understand the behavior of quantum Markov chains arises from the fact that quantum mechanical operators do not commute in general. As a result we start by explaining two techniques of how to deal with non-commuting matrices: the spectral pinching method and complex interpolation theory. Once the reader is familiar with these techniques a novel inequality is presented that extends the celebrated Golden-Thompson inequality to arbitrarily many matrices. This inequality is the key ingredient in understanding approximate quantum Markov chains and it answers a question from matrix analysis that was open since 1973, i.e., if Lieb's triple matrix inequality can be extended to more than three matrices. Finally, we carefully discuss the properties of approximate quantum Markov chains and their implications.
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