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 Mat. Sb., 2016, Volume 207, Number 5, Pages 93–142 (Mi msb8561)

Measures of correlations in infinite-dimensional quantum systems

M. E. Shirokov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Several important measures of correlations of the state of a finite-dimensional composite quantum system are defined as linear combinations of marginal entropies of this state. This paper is devoted to infinite-dimensional generalizations of such quantities and to an analysis of their properties.
We introduce the notion of faithful extension of a linear combination of marginal entropies and consider several concrete examples, the simplest of which are quantum mutual information and quantum conditional entropy. Then we show that quantum conditional mutual information can be defined uniquely 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 correlations in multipartite quantum systems are also considered. Applications of the results to the theory of infinite-dimensional quantum channels and their capacities are considered. The existence of a Fawzi-Renner recovery channel reproducing marginal states for all tripartite states (including states with infinite marginal entropies) is shown.
Bibliography: 47 titles.

Keywords: von Neumann entropy, marginal entropy, quantum mutual information, quantum channel, entanglement-assisted capacity.

 Funding Agency Grant Number Russian Science Foundation 14-21-00162 The research was funded by the grant from the Russian Science Foundation (project no. 14-21-00162).

DOI: https://doi.org/10.4213/sm8561

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English version:
Sbornik: Mathematics, 2016, 207:5, 724–768

Bibliographic databases:

UDC: 519.248.3
MSC: Primary 81P45; Secondary 46N50

Citation: M. E. Shirokov, “Measures of correlations in infinite-dimensional quantum systems”, Mat. Sb., 207:5 (2016), 93–142; Sb. Math., 207:5 (2016), 724–768

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8561
• https://doi.org/10.4213/sm8561
• http://mi.mathnet.ru/eng/msb/v207/i5/p93

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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 lower semicontinuity of the entropic disturbance and its applications in quantum information theory”, Izv. Math., 81:5 (2017), 1044–1060
2. M. E. Shirokov, “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, 22 pp.
3. M. E. Shirokov, “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
4. K. Sharma, M. M. Wilde, S. Adhikari, M. Takeoka, “Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels”, New J. Phys., 20 (2018), 063025
5. N. Davis, M. E. Shirokov, M. M. Wilde, “Energy-constrained two-way assisted private and quantum capacities of quantum channels”, Phys. Rev. A, 97:6 (2018), 062310
6. M. Junge, R. Renner, D. Sutter, M. M. Wilde, A. Winter, “Universal recovery maps and approximate sufficiency of quantum relative entropy”, Ann. Henri Poincaré, 19:10 (2018), 2955–2978
7. M. E. Shirokov, “Uniform finite-dimensional approximation of basic capacities of energy-constrained channels”, Quantum Inf. Process., 17:12 (2018), 322, 29 pp.
8. I. Ya. Aref'eva, I. V. Volovich, O. V. Inozemtsev, “Evolution of holographic entropy quantities for composite quantum systems”, Theoret. and Math. Phys., 197:3 (2018), 1838–1844
9. M. M. Wilde, H. Qi, “Energy-constrained private and quantum capacities of quantum channels”, IEEE Trans. Inform. Theory, 64:12 (2018), 7802–7827
10. M. E. Shirokov, “Uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy”, J. Phys. A, 52:1 (2019), 014001, 31 pp.
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