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Teor. Veroyatnost. i Primenen., 2007, Volume 52, Issue 2, Pages 301–335 (Mi tvp174)  

This article is cited in 5 scientific papers (total in 5 papers)

On properties of quantum channels related to their classical capacity

M. E. Shirokov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper is devoted to further study of the Holevo capacity of infinite-dimensional quantum channels. The existence of a unique optimal average state for a quantum channel constrained by an arbitrary convex set of states is shown. The minimax expression for the Holevo capacity of a constrained channel is obtained. The $\chi$-function and the convex closure of the output entropy of an infinite-dimensional quantum channel are considered. It is shown that the $\chi$-function of an arbitrary channel is lower semicontinuous on the set of all states and has continuous restrictions to subsets of states with continuous output entropy. The explicit expression for the convex closure of the output entropy of an infinite-dimensional quantum channel is obtained and its properties are explored. It is shown that the convex closure of the output entropy coincides with the convex hull of the output entropy on the set of states with finite output entropy and, similarly to the $\chi$-function, it has continuous restrictions to subsets of states with continuous output entropy. The applications of the obtained results to the theory of entanglement are considered. The properties of the convex closure of the output entropy make it possible to generalize some results related to the additivity problem to the infinite-dimensional case.

Keywords: quantum state, entropy, quantum channel, the Holevo capacity, the $\chi$-function, convex closure of the output entropy of a quantum channel.

DOI: https://doi.org/10.4213/tvp174

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English version:
Theory of Probability and its Applications, 2008, 52:2, 250–276

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Received: 04.09.2005

Citation: M. E. Shirokov, “On properties of quantum channels related to their classical capacity”, Teor. Veroyatnost. i Primenen., 52:2 (2007), 301–335; Theory Probab. Appl., 52:2 (2008), 250–276

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. E. Shirokov, A. S. Holevo, “On Approximation of Infinite-Dimensional Quantum Channels”, Problems Inform. Transmission, 44:2 (2008), 73–90  mathnet  crossref  mathscinet  isi  elib
    2. M. E. Shirokov, “On Channels with Finite Holevo Capacity”, Theory Probab. Appl., 53:4 (2009), 648–662  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. M. E. Shirokov, “On properties of the space of quantum states and their application to the construction of entanglement monotones”, Izv. Math., 74:4 (2010), 849–882  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. A. S. Holevo, “Gaussian optimizers and the additivity problem in quantum information theory”, Russian Math. Surveys, 70:2 (2015), 331–367  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Chanda T., Das T., Mal Sh., Sen(De) Aditi, Sen U., “Canonical Leggett-Garg Inequality: Nonclassicality of Temporal Quantum Correlations Under Energy Constraint”, Phys. Rev. A, 98:2 (2018), 022138  crossref  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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