Relative Entropy via Distribution of Observables

IF 0.6 4区数学 Q4 MATHEMATICS, APPLIED

Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-03-03 DOI:10.1142/s0219025723500212

G. Androulakis, Tiju Cherian John

引用次数: 2

Abstract

We obtain formulas for Petz-R\'enyi and Umegaki relative entropy from the idea of distribution of a positive selfadjoint operator. Classical results on R\'enyi and Kullback-Leibler divergences are applied to obtain new results and new proofs for some known results about Petz-R\'enyi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz-R\'enyi $\alpha$-relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory.

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通过观测分布的相对熵

从正自伴随算子的分布思想出发，得到了Petz-R\ enyi和Umegaki相对熵的表达式。应用经典的R′enyi散度和Kullback-Leibler散度的结果，得到了关于Petz-R′enyi和Umegaki相对熵的一些已知结果的新结果和新证明。其中最重要的是Petz-R\ enyi $\alpha$-相对熵有限的一个充分必要条件。这里给出的所有结果在有限和无限维度上都是有效的。这些结果对Fock空间中的态是有效的，因此适用于连续变量量子信息理论。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Infinite Dimensional Analysis Quantum Probability and Related Topics 数学-统计学与概率论

CiteScore

1.50

自引率

11.10%

发文量

审稿时长

>12 weeks

期刊介绍： In the past few years the fields of infinite dimensional analysis and quantum probability have undergone increasingly significant developments and have found many new applications, in particular, to classical probability and to different branches of physics. The number of first-class papers in these fields has grown at the same rate. This is currently the only journal which is devoted to these fields. It constitutes an essential and central point of reference for the large number of mathematicians, mathematical physicists and other scientists who have been drawn into these areas. Both fields have strong interdisciplinary nature, with deep connection to, for example, classical probability, stochastic analysis, mathematical physics, operator algebras, irreversibility, ergodic theory and dynamical systems, quantum groups, classical and quantum stochastic geometry, quantum chaos, Dirichlet forms, harmonic analysis, quantum measurement, quantum computer, etc. The journal reflects this interdisciplinarity and welcomes high quality papers in all such related fields, particularly those which reveal connections with the main fields of this journal.