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TMF, 2013, Volume 174, Number 2, Pages 331–341 (Mi tmf8389)  

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

Extreme bosonic linear channels

A. S. Holevo

Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia

Abstract: The set of all channels with a fixed input and output is convex. We first give a convenient formulation of the necessary and sufficient condition for a channel to be an extreme point of this set in terms of the complementary channel, a notion of great importance in quantum information theory. This formulation is based on the general approach to extremality of completely positive maps in an operator algebra in the spirit of Arveson. We then use this formulation to prove our main result: under certain nondegeneracy conditions, environmental purity is necessary and sufficient for the extremality of a bosonic linear (quasifree) channel. It hence follows that a Gaussian channel between finite-mode bosonic systems is extreme if and only if it has minimum noise.

Keywords: quantum information theory, bosonic linear channel, Gaussian channel, extremal channel, minimum noise

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00319_а
Russian Quantum Center


DOI: https://doi.org/10.4213/tmf8389

Full text: PDF file (419 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2013, 174:2, 288–297

Bibliographic databases:

Document Type: Article
Received: 18.06.2012

Citation: A. S. Holevo, “Extreme bosonic linear channels”, TMF, 174:2 (2013), 331–341; Theoret. and Math. Phys., 174:2 (2013), 288–297

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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, “Reversibility of a quantum channel: general conditions and their applications to bosonic linear channels”, J. Math. Phys., 54:11 (2013), 112201  crossref  mathscinet  zmath  adsnasa  isi
    2. A. S. Holevo, “Gaussian classical-quantum channels: gain from entanglement-assistance”, Problems Inform. Transmission, 50:1 (2014), 1–14  mathnet  crossref  mathscinet  isi
    3. J.-P. Pellonpaa, “Modules and extremal completely positive maps”, Positivity, 18:1 (2014), 61–79  crossref  mathscinet  zmath  isi  elib
    4. A. S. Holevo, “Gaussian optimizers and the additivity problem in quantum information theory”, Russian Math. Surveys, 70:2 (2015), 331–367  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. K. K. Sabapathy, “Quantum-optical channels that output only classical states”, Phys. Rev. A, 92:5 (2015), 052301  crossref  adsnasa  isi  elib
    6. D.-Sh. Wang, “_orig convex decomposition of dimension-altering quantum channels”, Int. J. Quantum Inf., 14:8 (2016), 1650045  crossref  mathscinet  zmath  isi  scopus
    7. J. S. Ivan, K. K. Sabapathy, R. Simon, “Scaling maps of s-ordered quasiprobabilities are either nonpositive or completely positive”, Phys. Rev. A, 96:2 (2017), 022114  crossref  isi
    8. Lami L., Sabapathy K.K., Winter A., “All Phase-Space Linear Bosonic Channels Are Approximately Gaussian Dilatable”, New J. Phys., 20 (2018), 113012  crossref  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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