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Izv. RAN. Ser. Mat., 2006, Volume 70, Issue 6, Pages 193–222 (Mi izv730)  

This article is cited in 19 scientific papers (total in 19 papers)

Entropy characteristics of subsets of states. I

M. E. Shirokov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We study the properties of quantum entropy and $\chi$-capacity (regarded as a function of sets of quantum states) in the infinite-dimensional case. We obtain conditions for the boundedness and continuity of the restriction of the entropy to a subset of quantum states, as well as conditions for the existence of the state with maximal entropy in certain subsets. The notion of $\chi$-capacity is considered for an arbitrary subset of states. The existence of an optimal average is proved for an arbitrary subset with finite $\chi$-capacity. We obtain a sufficient condition for the existence of an optimal measure and prove a generalized maximal distance property.

DOI: https://doi.org/10.4213/im730

Full text: PDF file (767 kB)
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English version:
Izvestiya: Mathematics, 2006, 70:6, 1265–1292

Bibliographic databases:

UDC: 519.722
MSC: 62B10, 81P68, 94A40
Received: 23.12.2005

Citation: M. E. Shirokov, “Entropy characteristics of subsets of states. I”, Izv. RAN. Ser. Mat., 70:6 (2006), 193–222; Izv. Math., 70:6 (2006), 1265–1292

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. 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
    2. M. E. Shirokov, “Entropy characteristics of subsets of states. II”, Izv. Math., 71:1 (2007), 181–218  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. 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
    4. M. E. Shirokov, “A Property of the Output Entropy of a Positive Map of Spaces of Nuclear Operators”, Math. Notes, 87:3 (2010), 449–451  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Shirokov M.E., “Continuity of the von Neumann Entropy”, Comm. Math. Phys., 296:3 (2010), 625–654  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. 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
    7. A. S. Holevo, “Entropy gain and the Choi–Jamiolkowski correspondence for infinite-dimensional quantum evolutions”, Theoret. and Math. Phys., 166:1 (2011), 123–138  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    8. M. E. Shirokov, “The continuity of the output entropy of positive maps”, Sb. Math., 202:10 (2011), 1537–1564  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. Holevo A.S., “The entropy gain of quantum channels”, IEEE International Symposium on Information Theory Proceedings (ISIT), 2011, 289–292  crossref  isi  scopus
    10. Weis Stephan, “Continuity of the Maximum-Entropy Inference”, Commun. Math. Phys, 2014  crossref  mathscinet  scopus
    11. Weis S., Knauf A., Ay N., Zhao M.-J., “Maximizing the Divergence From a Hierarchical Model of Quantum States”, Open Syst. Inf. Dyn., 22:1 (2015), 1550006  crossref  mathscinet  zmath  isi  scopus
    12. M. E. Shirokov, “Measures of correlations in infinite-dimensional quantum systems”, Sb. Math., 207:5 (2016), 724–768  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. 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
    14. Shirokov M.E., “Squashed entanglement in infinite dimensions”, J. Math. Phys., 57:3 (2016), 032203  crossref  mathscinet  zmath  isi  elib  scopus
    15. 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
    16. Shirokov M.E., “Adaptation of the Alicki-Fannes-Winter Method For the Set of States With Bounded Energy and Its Use”, Rep. Math. Phys., 81:1 (2018), 81–104  crossref  mathscinet  isi  scopus
    17. 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  mathscinet  isi  elib
    18. Wilde M.M., “Entanglement Cost and Quantum Channel Simulation”, Phys. Rev. A, 98:4 (2018), 042338  crossref  mathscinet  isi  scopus
    19. M. E. Shirokov, “Upper bounds for the Holevo quantity and their use”, Problems Inform. Transmission, 55:3 (2019), 201–217  mathnet  crossref  crossref  mathscinet  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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