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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 4, Pages 189–224 (Mi izv2815)  

This article is cited in 8 scientific papers (total in 8 papers)

On properties of the space of quantum states and their application to the construction of entanglement monotones

M. E. Shirokov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider infinite-dimensional versions of the notions of the convex hull and convex roof of a function defined on the set of quantum states. We obtain sufficient conditions for the coincidence and continuity of restrictions of different convex hulls of a given lower semicontinuous function to the subset of states with bounded mean generalized energy (an affine lower semicontinuous non-negative function). These results are used to justify an infinite-dimensional generalization of the convex roof construction of entanglement monotones that is widely used in finite dimensions. We give several examples of entanglement monotones produced by the generalized convex roof construction. In particular, we consider an infinite-dimensional generalization of the notion of Entanglement of Formation and study its properties.

Keywords: convex hull and convex roof of a function, quantum state, entanglement monotone, entanglement of formation.

DOI: https://doi.org/10.4213/im2815

Full text: PDF file (791 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2010, 74:4, 849–882

Bibliographic databases:

UDC: 519.248.3
MSC: 46N50, 81P40
Received: 16.06.2008
Revised: 21.04.2009

Citation: M. E. Shirokov, “On properties of the space of quantum states and their application to the construction of entanglement monotones”, Izv. RAN. Ser. Mat., 74:4 (2010), 189–224; Izv. Math., 74:4 (2010), 849–882

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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, “Schmidt Number and Partially Entanglement-Breaking Channels in Infinite-Dimensional Quantum Systems”, Math. Notes, 93:5 (2013), 766–779  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. 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
    3. Chang M., Quantum Stochastics, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge Univ Press, 2015  crossref  mathscinet  zmath  isi  scopus
    4. M. E. Shirokov, “Estimates for discontinuity jumps of information characteristics of quantum systems and channels”, Problems of Information Transmission, 52:3 (2016), 239–264  mathnet  crossref  mathscinet  isi  elib
    5. Shirokov M.E., “Squashed entanglement in infinite dimensions”, J. Math. Phys., 57:3 (2016), 032203  crossref  mathscinet  zmath  isi  elib  scopus
    6. Regula B., “Convex Geometry of Quantum Resource Quantification”, J. Phys. A-Math. Theor., 51:4 (2018), 045303  crossref  mathscinet  zmath  isi  scopus
    7. Wilde M.M., “Entanglement Cost and Quantum Channel Simulation”, Phys. Rev. A, 98:4 (2018), 042338  crossref  mathscinet  isi
    8. Sakai Yu., “Generalized Fano-Type Inequality For Countably Infinite Systems With List-Decoding”, Proceedings of 2018 International Symposium on Information Theory and Its Applications (Isita2018), IEEE, 2018, 727–731  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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