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

arXiv:1602.05930 · Problems of Information Transmission

Estimates for discontinuity jumps of information characteristics of quantum systems and channels

Максим Евгеньевич Широков

Published Jul 1, 2016• 4 citations
arXivHTMLPDF

Abstract

Quantitative analysis of discontinuity of information characteristics of quantum states and channels is presented. Estimates for discontinuity jump (loss) of the von Neumann entropy for a given converging sequence of states are obtained. It is shown, in particular, that for any sequence the loss of entropy is upper bounded by the loss of mean energy (with the coefficient characterizing the Hamiltonian of a system). Then we prove that discontinuity jumps of basic measures of classical and quantum correlations in composite quantum systems are upper bounded by the loss of one of the marginal entropies (with a corresponding coefficient). Quantitative discontinuity analysis of the output entropy of a quantum operation and of basic information characteristics of a quantum channel considered as functions of a pair (channel, input state) is presented.

DOI
10.1134/s0032946016030030

Related papers

Notes on entropic characteristics of quantum channels

Alexey E. Rastegin

Abstract One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss a few channel characteristics expressed by means of generalized entropies. Such characteristics can often be treated in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the q-average output entropy of degree q ≥ 1 is bounded from above by the q-entropy of the input density matrix. The concavity properties of the (q, s)-entropy exchange are considered. Fano type quantum bounds on the (q, s)-entropy exchange are derived. We also give upper bounds on the map (q, s)-entropies in terms of the output entropy, corresponding to the completely mixed input.

arXiv:1206.3056•Open Physics•Oct 24, 2012•4 citations

Uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy

M E Shirokov

Abstract We obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension (if it is finite) and on the input energy bound (if the input dimension is infinite). We pay special attention to the case in which a multimode quantum oscillator is an input system. First, we prove continuity bounds for the output conditional mutual information for a single channel and for copies of a channel. Then we obtain estimates for the variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. As a result, tight and close-to-tight continuity bounds for basic capacities of quantum channels, which depend on the input dimension, are obtained. They complement the Leung–Smith continuity bounds, which depend on the output dimension. Finally, we obtain tight and close-to-tight continuity bounds for basic capacities of infinite-dimensional energy-constrained channels with respect to the energy-constrained Bures distance generating the strong convergence of quantum channels.

arXiv:1604.00568•Journal of Physics A Mathematical and Theoretical•Oct 26, 2018•35 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

Relating Entropies of Quantum Channels

Dariusz Kurzyk, Łukasz Pawela, Zbigniew Puchała

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.

arXiv:2106.09629•Entropy•Aug 10, 2021•3 citations

On average output entropy of a quantum channel

Максим Евгеньевич Широков

We describe analytical properties of the average output entropy of a quantum channel as a function of a pair (channel, input ensemble). In particular, tight semicontinuity bounds for this function with the rank/energy constraints are obtained by using the modified semicontinuity bounds for the quantum conditional entropy of quantum-classical states and a special approximation technique. Several applications are considered. New semicontinuity and continuity bounds for the output Holevo information of a channel as a function of a pair (channel, input ensemble) are obtained. The semicontinuity bound for the entanglement of formation with the rank constraint obtained in [1] is improved. In the preliminary part, some results concerning ensembles of quantum states are presented. In particular, a new useful metric on the set of generalized ensembles is proposed and explored. The concept of passive energy of an ensemble introduced here plays an important role in the article.

arXiv:2404.07978•Apr 11, 2024•0 citations

Strong convergence of quantum channels: Continuity of the Stinespring dilation and discontinuity of the unitary dilation

M. E. Shirokov

It is shown that a sequence {Φn} of quantum channels strongly converges to a quantum channel Φ0 if and only if there exist a common environment for all the channels and a corresponding sequence {Vn} of Stinespring isometries strongly converging to a Stinespring isometry V0 of the channel Φ0. A quantitative description of the above characterization of the strong convergence in terms of appropriate metrics on the sets of quantum channels and Stinespring isometries is given. As a result, the uniform selective continuity of the complementary operation with respect to the strong convergence is established. The discontinuity of the unitary dilation is shown by constructing a strongly converging sequence of quantum channels that cannot be represented as a reduction of a strongly converging sequence of unitary channels. The Stinespring representation of strongly converging sequences of quantum channels allows us to prove the lower semicontinuity of the entropic disturbance as a function of a pair (channel, input ensemble). Some corollaries of this property are considered.

arXiv:1712.03219•Journal of Mathematical Physics•Aug 1, 2020•23 citations
Quantitative analysis of discontinuity of information characteristics of quantum states and channels is presented. Estimates for discontinuity jump (loss) of the von Neumann entropy for a given converging sequence of states are obtained. It is shown, in particular, that for any sequence the loss of entropy is upper bounded by the loss of mean energy (with the coefficient characterizing the Hamiltonian of a system). Then we prove that discontinuity jumps of basic measures of classical and quantum correlations in composite quantum systems are upper bounded by the loss of one of the marginal entropies (with a corresponding coefficient). Quantitative discontinuity analysis of the output entropy of a quantum operation and of basic information characteristics of a quantum channel considered as functions of a pair (channel, input state) is presented.
Abstract One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss a few channel characteristics expressed by means of generalized entropies. Such characteristics can often be treated in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the q-average output entropy of degree q ≥ 1 is bounded from above by the q-entropy of the input density matrix. The concavity properties of the (q, s)-entropy exchange are considered. Fano type quantum bounds on the (q, s)-entropy exchange are derived. We also give upper bounds on the map (q, s)-entropies in terms of the output entropy, corresponding to the completely mixed input.
Abstract We obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension (if it is finite) and on the input energy bound (if the input dimension is infinite). We pay special attention to the case in which a multimode quantum oscillator is an input system. First, we prove continuity bounds for the output conditional mutual information for a single channel and for copies of a channel. Then we obtain estimates for the variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. As a result, tight and close-to-tight continuity bounds for basic capacities of quantum channels, which depend on the input dimension, are obtained. They complement the Leung–Smith continuity bounds, which depend on the output dimension. Finally, we obtain tight and close-to-tight continuity bounds for basic capacities of infinite-dimensional energy-constrained channels with respect to the energy-constrained Bures distance generating the strong convergence of quantum channels.
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.
In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.
We describe analytical properties of the average output entropy of a quantum channel as a function of a pair (channel, input ensemble). In particular, tight semicontinuity bounds for this function with the rank/energy constraints are obtained by using the modified semicontinuity bounds for the quantum conditional entropy of quantum-classical states and a special approximation technique. Several applications are considered. New semicontinuity and continuity bounds for the output Holevo information of a channel as a function of a pair (channel, input ensemble) are obtained. The semicontinuity bound for the entanglement of formation with the rank constraint obtained in [1] is improved. In the preliminary part, some results concerning ensembles of quantum states are presented. In particular, a new useful metric on the set of generalized ensembles is proposed and explored. The concept of passive energy of an ensemble introduced here plays an important role in the article.
It is shown that a sequence {Φn} of quantum channels strongly converges to a quantum channel Φ0 if and only if there exist a common environment for all the channels and a corresponding sequence {Vn} of Stinespring isometries strongly converging to a Stinespring isometry V0 of the channel Φ0. A quantitative description of the above characterization of the strong convergence in terms of appropriate metrics on the sets of quantum channels and Stinespring isometries is given. As a result, the uniform selective continuity of the complementary operation with respect to the strong convergence is established. The discontinuity of the unitary dilation is shown by constructing a strongly converging sequence of quantum channels that cannot be represented as a reduction of a strongly converging sequence of unitary channels. The Stinespring representation of strongly converging sequences of quantum channels allows us to prove the lower semicontinuity of the entropic disturbance as a function of a pair (channel, input ensemble). Some corollaries of this property are considered.