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首页> 外文期刊>Annales Henri Poincare >Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules
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Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

机译：Quantum Markov链：复发，Schur函数和分裂规则

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In this work, we study the recurrence problem for quantum Markov chains, which are quantum versions of classical Markov chains introduced by S. Gudder and described in terms of completely positive maps. A notion of monitored recurrence for quantum Markov chains is examined in association with Schur functions, which codify information on the first return to some given state or subspace. Such objects possess important factorization and decomposition properties which allow us to obtain probabilistic results based solely on those parts of the graph where the dynamics takes place, the so-called splitting rules. These rules also yield an alternative to the folding trick to transform a doubly infinite system into a semi-infinite one which doubles the number of internal degrees of freedom. The generalization of Schur functions-so-called FR-functions-to the general context of closed operators in Banach spaces is the key for the present applications to open quantum systems. An important class of examples included in this setting are the open quantum random walks, as described by S. Attal et al., but we will state results in terms of general completely positive trace-preserving maps. We also take the opportunity to discuss basic results on recurrence of finite- dimensional iterated quantum channels and quantum versions of Kac's Lemma, in close association with recent results on the subject.
机译：在这项工作中，我们研究了Quantum Markov链的复发问题，这是S.普通型马尔可夫链的量子版本的古典马尔可夫链。与Schur函数相关联检查量子马尔可夫链的监测复发的概念，该职能将关于第一次返回的信息编纂给一些给定的状态或子空间。这些物体具有重要的分解和分解特性，其允许我们仅基于所谓的拆分规则来获得基于图表的图表的那些部分来获得概率结果。这些规则还产生了折叠技巧的替代方案，以将双重无限系统转换为半无限制，这使得内部自由度的数量加倍。 Schur函数的概念 - 所谓的FR函数 - Banach空间中封闭运营商的一般背景是本应用打开量子系统的关键。如S. Attal等人所描述的，该设置中包含的一个重要类别是开放量子随机漫步，但我们将在一般完全正追踪地图方面说明结果。我们还借此机会讨论基本结果对有限尺寸迭代量子通道和量子版本的KAC引理的量子型，与近期对象的结果密切相关。

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