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arXiv:2201.11477 · Physica Scripta

Quantifying continuity of characteristics of composite quantum systems

Максим Евгеньевич Широков

Published Mar 6, 2023• 11 citations
arXivHTMLPDF

Abstract

Abstract We describe the most general form of the Alicki–Fannes–Winter method for obtaining uniform continuity bounds for characteristics of composite quantum systems and consider several modifications of this method, which make it applicable to a wide class of characteristics in both finite-dimensional and infinite-dimensional cases. We present uniform continuity bounds for the most important characteristics of composite quantum systems. Along with the results obtained earlier by various authors, we describe a number of new results proved by the proposed methods. In particular, we obtain new continuity bounds for the quantum discord, one-way classical correlation and its regularization in finite-dimensional and infinite-dimensional bipartite quantum systems.

DOI
10.1088/1402-4896/acc1b3

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We consider methods for obtaining local lower bounds on characteristics of quantum (resp. classical) systems, i.e., lower bounds valid in the trace norm ϵ\epsilonϵ -neighborhood of a given state (resp. probability distribution). Our basic tool is the quasi-classical modification of the Alicki–Fannes–Winter technique which gives faithful one-side continuity bounds with the rank/energy constraint imposed on only one of the two quantum states (resp. probability distributions). We also propose a new universal method that allows us to obtain easy computable faithful local lower bounds on many important characteristics of quantum (resp. classical) systems. The main attention is paid to infinite-dimensional systems.
An universal approximation technique for analysis of different characteristics of states of composite infinite-dimensional quantum systems is proposed and used to prove general results concerning the properties of correlation and entanglement measures in such systems. Then these results are applied to the study of three important characteristics: the relative entropy of πππ-entanglement, the Rains bound (the unregularized and regularized versions of both characteristics are considered) and the conditional entanglement of mutual information. In particular, we analyse continuity and convexity properties of the above entanglement measures, prove several results simplifying their definitions and establish a finite-dimensional approximation property for these characteristics that allows us to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.
A universal approximation technique for analysis of different characteristics of states of composite infinite-dimensional quantum systems is proposed and used to prove general results on the properties of correlation and entanglement measures in such systems. Then these results are applied to the study of three important characteristics: the relative entropy of π-entanglement, the Rains bound (the unregularized and regularized versions of both characteristics are considered) and the conditional entanglement of mutual information. In particular, we analyze continuity and convexity properties of the above entanglement measures, prove several results simplifying their definitions and establish a finite-dimensional approximation property for these characteristics that allows us to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.
We consider universal methods for obtaining (uniform) continuity bounds for characteristics of multipartite quantum systems. We pay special attention to infinite-dimensional multipartite quantum systems under the energy constraints. By these methods, we obtain continuity bounds for several important characteristics of a multipartite quantum state: the quantum (conditional) mutual information, the squashed entanglement, the c-squashed entanglement, and the conditional entanglement of mutual information. The continuity bounds for the multipartite quantum mutual information are asymptotically tight for large dimension/energy. The obtained results are used to prove the asymptotic continuity of the n-partite squashed entanglement, the c-squashed entanglement, and the conditional entanglement of mutual information under the energy constraints.
Abstract We describe an advanced version of the AFW-technique proposed in [M. E. Shirokov, ‘‘Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki–Fannes–Winter technique’’, Lett. Math. Phys. 113, 121 (2023); M. E. Shirokov, ‘‘Local lower bounds on characteristics of quantum and classical systems’’, Lobachevskii J. Math. 44, 2169–2191 (2023)] which allows us to obtain lower semicontinuity bounds, continuity bounds and local lower bounds for characteristics of quantum systems and discrete random variables. We consider applications of the new version of the AFW-technique to several basic characteristics of quantum systems (the von Neumann entropy, the energy-type functionals, the quantum relative entropy, the conditional entropy and the entanglement of formation).