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 Mat. Sb., 2011, Volume 202, Number 10, Pages 131–160 (Mi msb7720)

The continuity of the output entropy of positive maps

M. E. Shirokov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Global and local continuity conditions for the output von Neumann entropy for positive maps between Banach spaces of trace-class operators in separable Hilbert spaces are obtained. Special attention is paid to completely positive maps: infinite dimensional quantum channels and operations.
It is shown that as a result of some specific properties of the von Neumann entropy (as a function on the set of density operators) several results on the output entropy of positive maps can be obtained, which cannot be derived from the general properties of entropy type functions. In particular, it is proved that global continuity of the output entropy of a positive map follows from its finiteness. A characterization of positive linear maps preserving continuity of the entropy (in the following sense: continuity of the entropy on an arbitrary subset of input operators implies continuity of the output entropy on this subset) is obtained. A connection between the local continuity properties of two completely positive complementary maps is considered.
Bibliography: 21 titles.

Keywords: von Neumann entropy, positive trace-class operator, quantum operation, quantum channel, $\mathrm{PCE}$-property of positive maps.

DOI: https://doi.org/10.4213/sm7720

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English version:
Sbornik: Mathematics, 2011, 202:10, 1537–1564

Bibliographic databases:

UDC: 519.248.3
MSC: 81R15, 94A17

Citation: M. E. Shirokov, “The continuity of the output entropy of positive maps”, Mat. Sb., 202:10 (2011), 131–160; Sb. Math., 202:10 (2011), 1537–1564

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7720
• https://doi.org/10.4213/sm7720
• http://mi.mathnet.ru/eng/msb/v202/i10/p131

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This publication is cited in the following articles:
1. Crann J., Neufang M., “Quantum Channels Arising From Abstract Harmonic Analysis”, J. Phys. A-Math. Theor., 46:4 (2013), 045308
2. M. E. Shirokov, “On characterization of positive maps preserving continuity of the von Neumann entropy”, Russian Math. Surveys, 71:5 (2016), 965–966
3. M. E. Shirokov, “Estimates for discontinuity jumps of information characteristics of quantum systems and channels”, Problems of Information Transmission, 52:3 (2016), 239–264
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