Majorization and additivity for multimode bosonic Gaussian channels. (English. Russian original) Zbl 1317.81047

Theor. Math. Phys. 182, No. 2, 284-293 (2015); translation from Teor. Mat. Fiz. 182, No. 2, 338-349 (2015).
Summary: We obtain a multimode extension of the majorization theorem for bosonic Gaussian channels, in particular, giving sufficient conditions under which the Glauber coherent states are the only minimizers for concave functionals of the output state of such a channel. We discuss direct implications of this multimode majorization for the positive solution of the famous additivity problem in the case of Gaussian channels. In particular, we prove the additivity of the output Rényi entropies of arbitrary order \(p > 1\). Finally, we present an alternative, more direct derivation of a majorization property of the Husimi function established by Lieb and Solovej.


81P45 Quantum information, communication, networks (quantum-theoretic aspects)
94A40 Channel models (including quantum) in information and communication theory
94A24 Coding theorems (Shannon theory)
Full Text: DOI arXiv


[1] V. Giovannetti, A. S. Holevo, and R. García-Patrón, “A solution of the Gaussian optimizer conjecture,” Commun. Math. Phys. (online first 2014)
[2] Mari, A; Giovannetti, V; Holevo, A S, No article title, Nature Commun., 5, 3826, (2014)
[3] Giovannetti, V; Lloyd, S, No article title, Phys. Rev. A (3), 69, 062307, (2004) · Zbl 1232.81013
[4] Lieb, E H; Solovej, J P, No article title, Acta Math., 212, 379-398, (2014) · Zbl 1298.81116
[5] Wehrl, A, No article title, Rev. Modern Phys., 50, 221-260, (1978)
[6] Lieb, E, No article title, Commun. Math. Phys., 62, 35-41, (1978) · Zbl 0385.60089
[7] A. S. Holevo, Quantum Systems, Channels, Information [in Russian], MTsNMO, Moscow (2010); English transl. (De Gruyter Stud. Math. Phys., Vol. 16), de Gruyter, Berlin (2012).
[8] Holevo, A S; Sohma, M; Hirota, O, No article title, Rep. Math. Phys., 46, 343-358, (2000) · Zbl 0974.81006
[9] Berezin, F A, No article title, Math. USSR-Izv., 6, 1117-1151, (1972) · Zbl 0259.47004
[10] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory [in Russian], IKI, Moscow (2003); English transl prev. ed., North-Holland, Amsterdam (1992).
[11] Berezin, F A, No article title, Math. USSR-Sb., 17, 269-277, (1972) · Zbl 0279.47006
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