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 Uspekhi Mat. Nauk, 2006, Volume 61, Issue 2(368), Pages 113–152 (Mi umn1709)

Multiplicativity of $p$-norms of completely positive maps and the additivity problem in quantum information theory

A. S. Holevo

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The additivity problem is one of the most profound mathematical problems of quantum information theory. From an analytical point of view it is closely related to the multiplicative property, with respect to tensor products, of norms of maps on operator spaces equipped with the Schatten norms (non-commutative analogue of $l_p$-norms). In this paper we survey the current state of the problem.

DOI: https://doi.org/10.4213/rm1709

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English version:
Russian Mathematical Surveys, 2006, 61:2, 301–339

Bibliographic databases:

Document Type: Article
UDC: 519.248.3
MSC: Primary 94A40, 81P68; Secondary 94A17, 47B10, 47B65

Citation: A. S. Holevo, “Multiplicativity of $p$-norms of completely positive maps and the additivity problem in quantum information theory”, Uspekhi Mat. Nauk, 61:2(368) (2006), 113–152; Russian Math. Surveys, 61:2 (2006), 301–339

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn1709
• https://doi.org/10.4213/rm1709
• http://mi.mathnet.ru/eng/umn/v61/i2/p113

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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, “Superadditivity of the convex closure of the output entropy of a quantum channel”, Russian Math. Surveys, 61:6 (2006), 1186–1188
2. Szarek S.J., “On Norms of Completely Positive Maps”, Topics in Operator Theory: Operators, Matrices and Analytic Functions, Operator Theory Advances and Applications, 1, 2010, 535–538
3. M.M.. Wilde, Andreas Winter, Dong Yang, “Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy”, Commun. Math. Phys, 2014
4. A. S. Holevo, “Gaussian optimizers and the additivity problem in quantum information theory”, Russian Math. Surveys, 70:2 (2015), 331–367
5. Winter A., “Weak Locking Capacity of Quantum Channels Can be Much Larger Than Private Capacity”, J. Cryptology, 30:1 (2017), 1–21
6. Kaur E., Wilde M.M., “Relative Entropy of Steering: on Its Definition and Properties”, J. Phys. A-Math. Theor., 50:46 (2017), 465301
7. De Palma G., Trevisan D., Giovannetti V., “Gaussian States Minimize the Output Entropy of One-Mode Quantum Gaussian Channels”, Phys. Rev. Lett., 118:16 (2017), 160503
8. Kaur E., Wilde M.M., “Amortized Entanglement of a Quantum Channel and Approximately Teleportation-Simulable Channels”, J. Phys. A-Math. Theor., 51:3 (2018), 035303
9. De Palma G., “The Wehrl Entropy Has Gaussian Optimizers”, Lett. Math. Phys., 108:1 (2018), 97–116
10. D. Ding, M. M. Wilde, “Strong converse for the feedback-assisted classical capacity of entanglement-breaking channels”, Problems Inform. Transmission, 54:1 (2018), 1–19
11. 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
12. De Palma G., Trevisan D., Giovannetti V., Ambrosio L., “Gaussian Optimizers For Entropic Inequalities in Quantum Information”, J. Math. Phys., 59:8 (2018), 081101
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