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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

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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

Citation in format AMSBIB
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This publication is cited in the following articles:
1. K. K. Sabapathy, “Quantum-optical channels that output only classical states”, Phys. Rev. A, 92:5 (2015), 052301
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
3. A. S. Holevo, “On the proof of the majorization theorem for quantum Gaussian channels”, Russian Math. Surveys, 71:3 (2016), 585–587
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
5. I. V. Volovich, S. V. Kozyrev, “Manipulation of states of a degenerate quantum system”, Proc. Steklov Inst. Math., 294 (2016), 241–251
6. G. De Palma, A. Mari, S. Lloyd, V. Giovannetti, “Passive states as optimal inputs for single-jump lossy quantum channels”, Phys. Rev. A, 93:6 (2016), 062328
7. G. De Palma, D. Trevisan, V. Giovannetti, “Passive states optimize the output of bosonic Gaussian quantum channels”, IEEE Trans. Inform. Theory, 62:5 (2016), 2895–2906
8. G. De Palma, D. Trevisan, V. Giovannetti, “Gaussian states minimize the output entropy of the one-mode quantum attenuator”, IEEE Trans. Inf. Theory, 63:1 (2017), 728–737
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
10. A. S. Kholevo, “On the classical capacity of a channel with stationary quantum Gaussian noise”, Theory Probab. Appl., 62:4 (2018), 534–551
11. A. S. Holevo, “On the quantum Gaussian optimizers conjecture in the case $q=p$”, Russian Math. Surveys, 72:6 (2017), 1177–1179
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.
13. R. L. Frank, E. H. Lieb, “Norms of quantum Gaussian multi-mode channels”, J. Math. Phys., 58:6 (2017), 062204, 7 pp.
14. G. De Palma, D. Trevisan, V. Giovannetti, “Gaussian states minimize the output entropy of one-mode quantum Gaussian channels”, Phys. Rev. Lett., 118:16 (2017), 160503, 5 pp.
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
16. G. De Palma, “The Wehrl entropy has Gaussian optimizers”, Lett. Math. Phys., 108:1 (2018), 97–116
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.
18. G. De Palma, D. Trevisan, “The conditional entropy power inequality for bosonic quantum systems”, Comm. Math. Phys., 360:2 (2018), 639–662
19. De Palma G., “Uncertainty Relations With Quantum Memory For the Wehrl Entropy”, Lett. Math. Phys., 108:9 (2018), 2139–2152
20. S. V. Kozyrev, “Quantum transport in degenerate systems”, Proc. Steklov Inst. Math., 301 (2018), 134–143
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
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
23. De Palma G., “The Entropy Power Inequality With Quantum Conditioning”, J. Phys. A-Math. Theor., 52:8 (2019), 08LT03
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