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Uspekhi Mat. Nauk, 2015, Volume 70, Issue 2(422), Pages 141–180 (Mi umn9634)  

This article is cited in 23 scientific papers (total in 23 papers)

Gaussian optimizers and the additivity problem in quantum information theory

A. S. Holevo

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: This paper surveys two remarkable analytical problems of quantum information theory. The main part is a detailed report on the recent (partial) solution of the quantum Gaussian optimizer problem which establishes an optimal property of Glauber's coherent states — a particular case of pure quantum Gaussian states. The notion of a quantum Gaussian channel is developed as a non-commutative generalization of an integral operator with Gaussian kernel, and it is shown that the coherent states, and under certain conditions only they, minimize a broad class of concave functionals of the output of a Gaussian channel. Thus, the output states corresponding to a Gaussian input are the ‘least chaotic’, majorizing all the other outputs. The solution, however, is essentially restricted to the gauge-invariant case where a distinguished complex structure plays a special role. Also discussed is the related well-known additivity conjecture, which was solved in principle in the negative some five years ago. This refers to the additivity or multiplicativity (with respect to tensor products of channels) of information quantities related to the classical capacity of a quantum channel, such as the $(1\to p)$-norms or the minimal von Neumann or Rényi output entropies. A remarkable corollary of the present solution of the quantum Gaussian optimizer problem is that these additivity properties, while not valid in general, do hold in the important and interesting class of gauge-covariant Gaussian channels.
Bibliography: 65 titles.

Keywords: completely positive map, canonical commutation relations, Gaussian state, coherent state, quantum Gaussian channel, gauge covariance, von Neumann entropy, channel capacity, majorization.

Funding Agency Grant Number
Russian Science Foundation 14-21-00162
This work is supported by the Russian Science Foundation under grant 14-21-00162.


DOI: https://doi.org/10.4213/rm9634

Full text: PDF file (859 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2015, 70:2, 331–367

Bibliographic databases:

Document Type: Article
UDC: 519.248.3+517.983.2
MSC: Primary 94A40; Secondary 81P45, 81P68
Received: 11.01.2015

Citation: A. S. Holevo, “Gaussian optimizers and the additivity problem in quantum information theory”, Uspekhi Mat. Nauk, 70:2(422) (2015), 141–180; Russian Math. Surveys, 70:2 (2015), 331–367

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    2. A. S. Holevo, M. E. Shirokov, “On the Gain of Entanglement Assistance in the Classical Capacity of Quantum Gaussian Channels”, Math. Notes, 97:6 (2015), 974–977  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. A. S. Holevo, “On the proof of the majorization theorem for quantum Gaussian channels”, Russian Math. Surveys, 71:3 (2016), 585–587  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. A. N. Pechen, “On the speed gradient method for generating unitary quantum operations for closed quantum systems”, Russian Math. Surveys, 71:3 (2016), 597–599  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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    9. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. A. S. Kholevo, “On the classical capacity of a channel with stationary quantum Gaussian noise”, Theory Probab. Appl., 62:4 (2018), 534–551  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    11. A. S. Holevo, “On the quantum Gaussian optimizers conjecture in the case $q=p$”, Russian Math. Surveys, 72:6 (2017), 1177–1179  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. S. V. Kozyrev, A. A. Mironov, A. E. Teretenkov, I. V. Volovich, “Flows in non-equilibrium quantum systems and quantum photosynthesis”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 20:4 (2017), 1750021, 19 pp.  crossref  mathscinet  zmath  isi  scopus
    13. R. L. Frank, E. H. Lieb, “Norms of quantum Gaussian multi-mode channels”, J. Math. Phys., 58:6 (2017), 062204, 7 pp.  crossref  mathscinet  zmath  isi  scopus
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    15. S. B. Korolev, K. S. Tikhonov, T. Yu. Golubeva, Yu. M. Golubev, “Clusters on the basis of bright multimode light in a mixed state”, Opt. Spectrosc., 123:3 (2017), 411–418  crossref  isi  scopus
    16. G. De Palma, “The Wehrl entropy has Gaussian optimizers”, Lett. Math. Phys., 108:1 (2018), 97–116  crossref  mathscinet  zmath  isi  scopus
    17. S. Huber, R. König, “Coherent state coding approaches the capacity of non-Gaussian bosonic channels”, J. Phys. A, 51:18 (2018), 184001, 20 pp.  crossref  mathscinet  zmath  isi  scopus
    18. G. De Palma, D. Trevisan, “The conditional entropy power inequality for bosonic quantum systems”, Comm. Math. Phys., 360:2 (2018), 639–662  crossref  mathscinet  zmath  isi  scopus
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    20. S. V. Kozyrev, “Quantum transport in degenerate systems”, Proc. Steklov Inst. Math., 301 (2018), 134–143  mathnet  crossref  crossref  isi  elib  elib
    21. De Palma G. Trevisan D. Giovannetti V., “The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have Gaussianmaximizers”, Ann. Henri Poincare, 19:10 (2018), 2919–2953  crossref  mathscinet  zmath  isi  scopus
    22. De Palma G. Trevisan D. Giovannetti V. Ambrosio L., “Gaussian Optimizers For Entropic Inequalities in Quantum Information”, J. Math. Phys., 59:8 (2018), 081101  crossref  mathscinet  zmath  isi  scopus
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