ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 3, pp. 201–218.
Pleiades Publishing, Inc., 2010.
Original Russian Text
A.S. Holevo, M.E. Shirokov, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 3, pp. 3–21.
Mutual and Coherent Information
for Inﬁnite-Dimensional Quantum Channels
A. S. Holevo and M. E. Shirokov
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow
Received January 28, 2010; in ﬁnal form, June 7, 2010
Abstract—The paper is devoted to the study of quantum mutual information and coherent
information, two important characteristics of a quantum communication channel. Appropriate
deﬁnitions of these quantities in the inﬁnite-dimensional case are given, and their properties
are studied in detail. Basic identities relating the quantum mutual information and coherent
information of a pair of complementary channels are proved. An unexpected continuity prop-
erty of the quantum mutual information and coherent information, following from the above
identities, is observed. An upper bound for the coherent information is obtained.
One of achievements in quantum information theory is the discovery of a number of important
entropy and information characteristics of quantum systems (see, e.g., [1, 2]). Some of them, such
as the χ-capacity and quantum mutual information, have direct classical analogs; others, such as
the coherent information and various entanglement measures, either do not have such analogs or
they are trivial.
Until recently, the main attention in quantum information theory was paid to ﬁnite-dimen-
sional systems, but in recent years considerable interest to inﬁnite-dimensional systems appeared:
a broad class, important for applications in quantum optics, is formed by Bosonic Gaussian sys-
tems [2, ch. 11]. Note that properties of the entropy and relative entropy were studied in great
detail, including the inﬁnite-dimensional case, in connection with quantum statistical mechanics;
see, e.g., [3–5]. A study of entropy and information characteristics of quantum communication chan-
nels from the general viewpoint of operator theory in a separable Hilbert space was undertaken
in [6, 7], where quantities related to the classical capacity—in the ﬁrst place, the χ-capacity—were
investigated. The present work is devoted to two other characteristics, namely, the quantum mu-
tual information and coherent information. The ﬁrst is closely related to the entanglement-assisted
classical capacity, while the second, to the quantum capacity of a channel. One of the authors’
goals was to give an appropriate deﬁnition of these quantities in the inﬁnite-dimensional case that
would not require any additional artiﬁcial assumptions. A diﬃculty, which has been overcome, was
in uncertainties in expressions containing diﬀerences of entropies, each of which can be inﬁnite in
Supported in part by the Russian Foundation for Basic Research, project no. 09-01-00424a, and Scientiﬁc
Program “Mathematical Control Theory” of the Russian Academy of Sciences.
Supported in part by the Analytical Departmental Target Program “Development of the Scientiﬁc Po-
tential of the Higher School,” project no. 2.1.1/500, Federal Target Program “Research and Educational
Personnel of Innovation Russia,” project no. 1.2.1, contract no. 938, and Russian Foundation for Basic
Research, project no. 10-01-00139a.