Concentration of quantum states from quantum functional and transportation cost inequalities
Concentration of quantum states from quantum functional and transportation cost inequalities
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DOI：
10.1063/1.5023210
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发表时间：
2019-01-01
影响因子：
1.3
通讯作者：
Datta, Nilanjana
中科院分区：
文献类型：
--
作者：
Rouze, Cambyse;Datta, Nilanjana
量子函数不等式（例如，对数Sobolev和Poincare不等式）在研究本原量子马尔可夫半群的行为中有着广泛的应用。这些不等式的经典对应物通过所谓的2阶运输成本不等式（TC2）相互关联。后一个不等式依赖于概率分布集合上的度量的概念，称为2阶Wasserstein距离。(TC2)这又意味着一阶运输成本不等式（TC 1）。在本文中，我们引入不等式（TC 1）和（TC 2）的量子推广，利用适当的量子版本的Wasserstein距离，最近定义的Carlen和Maas和其他定义我们。我们建立，这些不等式是相互关联的，量子修改的对数Sobolev和庞加莱不等式，在经典的情况下。我们还表明，这些不等式意味着一定的浓度型结果的不变状态的基础半群。以退极化半群为例，导出了任意有限维满秩量子态的浓度不等式。这些不等式，然后应用到推导出的错误概率上界发生在有限块长度的量子参数估计的设置。由AIP Publishing授权出版。
Quantum functional inequalities (e.g., the logarithmic Sobolev and Poincare inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these inequalities are related to each other via a so-called transportation cost inequality of order 2 (TC2). The latter inequality relies on the notion of a metric on the set of probability distributions called the Wasserstein distance of order 2. (TC2) in turn implies a transportation cost inequality of order 1 (TC1). In this paper, we introduce quantum generalizations of the inequalities (TC1) and (TC2), making use of appropriate quantum versions of the Wasserstein distances, one recently defined by Carlen and Maas and the other defined by us. We establish that these inequalities are related to each other, and to the quantum modified logarithmic Sobolev- and Poincare inequalities, as in the classical case. We also show that these inequalities imply certain concentration-type results for the invariant state of the underlying semigroup. We consider the example of the depolarizing semigroup to derive concentration inequalities for any finite dimensional full-rank quantum state. These inequalities are then applied to derive upper bounds on the error probabilities occurring in the setting of finite blocklength quantum parameter estimation. Published under license by AIP Publishing.