Entangled Markov chains
Entangled Markov chains
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DOI：
10.1007/s10231-004-0118-4
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发表时间：
2005-08-01
影响因子：
1
通讯作者：
Fidaleo, Francesco
中科院分区：
文献类型：
--
作者：
Accardi, Luigi;Fidaleo, Francesco
受寻找经典随机游动的一个满意的量子推广问题的启发，我们构造了一类新的量子马尔可夫链，它们同时是由相应的经典马尔可夫链纯生成和唯一确定的.我们认为，这一建设收益率作为一个推论，构造“纠缠”的经典随机游动的量子类似物问题的解决方案在一定的意义上，我们用Schur乘法代替通常的矩阵乘法，从相应的经典公式出发，得到了这些量子链的联合关联公式，阐明了Schur乘法与纠缠之间的联系，表明这些量子链是矢量态的极限，在给定的基（例如量子信息的计算基）中，其幅度是相应经典链的联合概率的复平方根。特别地，当在此基础上限制于投影者时，量子链还原为经典链。在这个意义上，我们谈到纠缠提升，量子的情况下，一个经典的马尔可夫链。由于随机游动是特殊的马尔可夫链，我们的一般结构也给出了一个解决方案的问题，激发我们的研究，考虑到可能的应用量子统计力学太多，我们证明了遍历类型的纠缠马尔可夫链与有限的状态空间（从而排除随机游动）是完全由相应的遍历类型的基础经典链。
Motivated by the problem of finding a satisfactory quantum generalization of the classical random walks, we construct a new class of quantum Markov chains which are at the same time purely generated and uniquely determined by a corresponding classical Markov chain. We argue that this construction yields as a corollary, a solution to the problem of constructing quantum analogues of classical random walks which are "entangled" in a sense specified in the paper.The formula giving the joint correlations of these quantum chains is obtained from the corresponding classical formula by replacing the usual matrix multiplication by Schur multiplication.The connection between Schur multiplication and entanglement is clarified by showing that these quantum chains are the limits of vector states whose amplitudes, in a given basis (e.g. the computational basis of quantum information), are complex square roots of the joint probabilities of the corresponding classical chains. In particular, when restricted to the projectors on this basis, the quantum chain reduces to the classical one. In this sense we speak of entangled lifting, to the quantum case, of a classical Markov chain. Since random walks are particular Markov chains, our general construction also gives a solution to the problem that motivated our study.In view of possible applications to quantum statistical mechanics too, we prove that the ergodic type of an entangled Markov chain with finite state space (thus excluding random walks) is completely determined by the corresponding ergodic type of the underlying classical chain.