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Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 1, Pages 98–114 (Mi tvp160)  

This article is cited in 34 scientific papers (total in 34 papers)

Continuous ensembles and the capacity of infinite-dimensional quantum channels

A. S. Holevo, M. E. Shirokov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper is devoted to the study of $\chi$-capacity, closely related to the classical capacity of infinite-dimensional quantum channels. For such channels generalized ensembles are defined as probability measures on the set of all quantum states. We establish the compactness of the set of generalized ensembles with averages in an arbitrary compact subset of states. This result enables us to obtain a sufficient condition for the existence of the optimal generalized ensemble for an infinite-dimensional channel with input constraint. This condition is shown to be fulfilled for Bosonic Gaussian channels with constrained mean energy. In the case of convex constraints, a characterization of the optimal generalized ensemble extending the “maximal distance property” is obtained.

Keywords: quantum channel, $\chi$-capacity, generalized ensemble.

DOI: https://doi.org/10.4213/tvp160

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English version:
Theory of Probability and its Applications, 2005, 50:1, 86–98

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Received: 21.09.2004

Citation: A. S. Holevo, M. E. Shirokov, “Continuous ensembles and the capacity of infinite-dimensional quantum channels”, Teor. Veroyatnost. i Primenen., 50:1 (2005), 98–114; Theory Probab. Appl., 50:1 (2005), 86–98

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. F. Werner, A. S. Holevo, M. E. Shirokov, “On the notion of entanglement in Hilbert spaces”, Russian Math. Surveys, 60:2 (2005), 359–360  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. M. E. Shirokov, “Entropy characteristics of subsets of states. I”, Izv. Math., 70:6 (2006), 1265–1292  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. A. S. Holevo, “Multiplicativity of $p$-norms of completely positive maps and the additivity problem in quantum information theory”, Russian Math. Surveys, 61:2 (2006), 301–339  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. M. E. Shirokov, “On the Strong CE-Property of Convex Sets”, Math. Notes, 82:3 (2007), 395–409  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. M. E. Shirokov, “On properties of quantum channels related to their classical capacity”, Theory Probab. Appl., 52:2 (2008), 250–276  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. Eisert J., Wolf M.M., “Gaussian quantum channels”, Quantum Information with Continous Variables of Atoms and Light, 2007, 23–42  crossref  mathscinet  zmath  isi  scopus
    7. 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
    8. A. S. Holevo, “Entanglement-Breaking Channels in Infinite Dimensions”, Problems Inform. Transmission, 44:3 (2008), 171–184  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    9. 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
    10. V. Yu. Protasov, M. E. Shirokov, “Generalized compactness in linear spaces and its applications”, Sb. Math., 200:5 (2009), 697–722  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Czekaj L., Horodecki P., “Purely quantum superadditivity of classical capacities of quantum multiple access channels”, Phys. Rev. Lett., 102:11 (2009), 110505, 4 pp.  crossref  adsnasa  isi  elib  scopus
    12. Shirokov M.E., “Continuity of the von Neumann Entropy”, Comm. Math. Phys., 296:3 (2010), 625–654  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    13. 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
    14. A. S. Holevo, M. E. Shirokov, “Mutual and coherent information for infinite-dimensional quantum channels”, Problems Inform. Transmission, 46:3 (2010), 201–218  mathnet  crossref  mathscinet  isi
    15. A. A. Kuznetsova, “Conditional entropy for infinite-dimensional quantum systems”, Theory Probab. Appl., 55:4 (2011), 709–717  mathnet  crossref  crossref  mathscinet  isi
    16. Barchielli A., Pellegrini C., “Jump-diffusion unravelling of a non-Markovian generalized Lindblad master equation”, J. Math. Phys., 51:11 (2010), 112104, 18 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    17. Wu Zh., Zhang Sh., Zhu Ch., “Quantum Jensen-Shannon Divergence Between Quantum Ensembles”, Appl. Math. Inf. Sci., 6:3 (2012), 509–514  mathscinet  isi  elib
    18. M. E. Shirokov, “Reversibility conditions for quantum channels and their applications”, Sb. Math., 204:8 (2013), 1215–1237  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    19. A. S. Holevo, M. E. Shirokov, “On classical capacities of infinite-dimensional quantum channels”, Problems Inform. Transmission, 49:1 (2013), 15–31  mathnet  crossref  isi
    20. M. E. Shirokov, “Criteria for equality in two entropic inequalities”, Sb. Math., 205:7 (2014), 1045–1068  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. Caruso F. Giovannetti V. Lupo C. Mancini S., “Quantum Channels and Memory Effects”, Rev. Mod. Phys., 86:4 (2014), 1203–1259  crossref  adsnasa  isi  scopus
    22. Giovannetti V. Garcia-Patron R. Cerf N.J. Holevo A.S., “Ultimate Classical Communication Rates of Quantum Optical Channels”, Nat. Photonics, 8:10 (2014), 796–800  crossref  adsnasa  isi  scopus
    23. A. S. Holevo, M. E. Shirokov, “Criterion of weak compactness for families of generalized quantum ensembles and its applications”, Theory Probab. Appl., 60:2 (2016), 320–325  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    24. Holevo A.S., “on the Constrained Classical Capacity of Infinite-Dimensional Covariant Quantum Channels”, J. Math. Phys., 57:1, SI (2016), 015203  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    25. Winter A., “Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints”, Commun. Math. Phys., 347:1 (2016), 291–313  crossref  mathscinet  zmath  isi  elib  scopus
    26. M. E. Shirokov, A. S. Holevo, “On lower semicontinuity of the entropic disturbance and its applications in quantum information theory”, Izv. Math., 81:5 (2017), 1044–1060  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    27. A. S. Kholevo, “On the classical capacity of a channel with stationary quantum Gaussian noise”, Theory Probab. Appl., 62:4 (2018), 534–551  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    28. Shirokov M.E., “Tight Uniform Continuity Bounds For the Quantum Conditional Mutual Information, For the Holevo Quantity, and For Capacities of Quantum Channels”, J. Math. Phys., 58:10 (2017), 102202  crossref  mathscinet  zmath  isi  scopus
    29. Davis N. Shirokov M.E. Wilde M.M., “Energy-Constrained Two-Way Assisted Private and Quantum Capacities of Quantum Channels”, Phys. Rev. A, 97:6 (2018), 062310  crossref  isi  scopus
    30. M. E. Shirokov, “On the energy-constrained diamond norm and its application in quantum information theory”, Problems Inform. Transmission, 54:1 (2018), 20–33  mathnet  crossref  isi  elib
    31. Filippov S.N., “Lower and Upper Bounds on Nonunital Qubit Channel Capacities”, Rep. Math. Phys., 82:2 (2018), 149–159  crossref  mathscinet  isi  scopus
    32. Wilde M.M., Qi H., “Energy-Constrained Private and Quantum Capacities of Quantum Channels”, IEEE Trans. Inf. Theory, 64:12 (2018), 7802–7827  crossref  mathscinet  zmath  isi  scopus
    33. Filippov S.N., “Evaluation of Non-Unital Qubit Channel Capacities”, Uchenye Zap. Kazan. Univ.-Ser. Fiz.-Mat. Nauki, 160:2 (2018), 258–265  mathnet  mathscinet  isi
    34. Shirokov M.E., “Uniform Continuity Bounds For Information Characteristics of Quantum Channels Depending on Input Dimension and on Input Energy”, J. Phys. A-Math. Theor., 52:1 (2019), 014001  crossref  mathscinet  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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