Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceedings of the Steklov Institute of Mathematics

On Local Continuity of Characteristics of Composite Quantum Systems

Максим Евгеньевич Широков

Published Mar 1, 2024• 1 citation
Publisher

Abstract

Abstract General methods of local continuity analysis of characteristics of infinite-dimensional composite quantum systems are considered. A new approximation technique for obtaining local continuity conditions for various characteristics of quantum systems is proposed and described in detail. This technique is used to prove several general results (a Simon-type dominated convergence theorem, a theorem on the preservation of continuity under convex mixtures, etc.). Local continuity conditions are derived for the following characteristics of composite quantum systems: the quantum conditional entropy, the quantum (conditional) mutual information, the one-way classical correlation and its regularization, the quantum discord and its regularization, the entanglement of formation and its regularization, and the constrained Holevo capacity of a partial trace and its regularization.

DOI
10.1134/s0081543824010206

Related papers

Approximation of composite quantum states: General results and applications

Максим Евгеньевич Широков

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.

Journal of Mathematical Physics•Dec 1, 2025•0 citations

Quantifying continuity of characteristics of composite quantum systems

Максим Евгеньевич Широков

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.

arXiv:2201.11477•Physica Scripta•Mar 6, 2023•11 citations

Approximation of multipartite quantum states: revised version with new applications

Максим Евгеньевич Широков

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 $π$-entanglement, the Rains bound (the unregularized and regularized versions of both characteristics are considered) and the conditional entanglement of mutual information. In particular, we 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.

arXiv:2401.02388•Jan 4, 2024•0 citations

Convergence Conditions for the Quantum Relative Entropy and Other Applications of the Deneralized Quantum Dini Lemma

Максим Евгеньевич Широков

Abstract We describe a generalized version of the result called quantum Dini lemma that was used previously for analysis of local continuity of basic correlation and entanglement measures. The generalization consists in considering sequences of functions instead of a single function. It allows to expand the scope of possible applications of the method. We prove two general dominated convergence theorems and the theorem about preserving local continuity under convex mixtures. By using these theorems we obtain several convergence conditions for the quantum relative entropy and for the mutual information of a quantum channel considered as a function of a pair (channel, input state).

arXiv:2205.09108•Lobachevskii Journal of Mathematics•Jul 1, 2022•6 citations

Uniform continuity bounds for characteristics of multipartite quantum systems

Максим Евгеньевич Широков

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.

arXiv:2007.00417•Journal of Mathematical Physics•Sep 1, 2021•7 citations

Measures of correlations in infinite-dimensional quantum systems

Максим Евгеньевич Широков

Several important measures of quantum correlations of a state of a finite-dimensional composite system are defined as linear combinations of marginal entropies of this state. This paper is devoted to the infinite-dimensional generalizations of such quantities and to the analysis of their properties. We introduce the notion of faithful extension of a linear combination of marginal entropies and consider several concrete examples starting with the quantum mutual information and the quantum conditional entropy. Then we show that the conditional mutual information can be uniquely defined as a lower semicontinuous function on the set of all states of a tripartite infinite-dimensional system possessing all the basic properties valid in finite dimensions. Infinite-dimensional generalizations of some other measures of quantum correlations in multipartite quantum systems are also considered. It is shown that almost all of these generalized measures are globally lower semicontinuous and possess local continuity properties which essentially simplify their use in analysis of quantum systems. In the second part of the paper we consider applications of the general results of its first part, in particular, to the theory of infinite-dimensional quantum channels and their capacities. We also show the existence of the Fawzi-Renner recovery channel reproducing marginal states for all tripartite states (including states with infinite marginal entropies) starting with the corresponding finite-dimensional result.

arXiv:1506.06377•Sbornik Mathematics•Mar 11, 2016•35 citations
Abstract General methods of local continuity analysis of characteristics of infinite-dimensional composite quantum systems are considered. A new approximation technique for obtaining local continuity conditions for various characteristics of quantum systems is proposed and described in detail. This technique is used to prove several general results (a Simon-type dominated convergence theorem, a theorem on the preservation of continuity under convex mixtures, etc.). Local continuity conditions are derived for the following characteristics of composite quantum systems: the quantum conditional entropy, the quantum (conditional) mutual information, the one-way classical correlation and its regularization, the quantum discord and its regularization, the entanglement of formation and its regularization, and the constrained Holevo capacity of a partial trace and its regularization.
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.
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.
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.
Abstract We describe a generalized version of the result called quantum Dini lemma that was used previously for analysis of local continuity of basic correlation and entanglement measures. The generalization consists in considering sequences of functions instead of a single function. It allows to expand the scope of possible applications of the method. We prove two general dominated convergence theorems and the theorem about preserving local continuity under convex mixtures. By using these theorems we obtain several convergence conditions for the quantum relative entropy and for the mutual information of a quantum channel considered as a function of a pair (channel, input state).
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.
Several important measures of quantum correlations of a state of a finite-dimensional composite system are defined as linear combinations of marginal entropies of this state. This paper is devoted to the infinite-dimensional generalizations of such quantities and to the analysis of their properties. We introduce the notion of faithful extension of a linear combination of marginal entropies and consider several concrete examples starting with the quantum mutual information and the quantum conditional entropy. Then we show that the conditional mutual information can be uniquely defined as a lower semicontinuous function on the set of all states of a tripartite infinite-dimensional system possessing all the basic properties valid in finite dimensions. Infinite-dimensional generalizations of some other measures of quantum correlations in multipartite quantum systems are also considered. It is shown that almost all of these generalized measures are globally lower semicontinuous and possess local continuity properties which essentially simplify their use in analysis of quantum systems. In the second part of the paper we consider applications of the general results of its first part, in particular, to the theory of infinite-dimensional quantum channels and their capacities. We also show the existence of the Fawzi-Renner recovery channel reproducing marginal states for all tripartite states (including states with infinite marginal entropies) starting with the corresponding finite-dimensional result.