ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 4, pp. 282–297.
Pleiades Publishing, Inc., 2006.
Original Russian Text
M.E. Shirokov, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 4, pp. 23–40.
On the Structure of Optimal Sets
for a Quantum Channel
M. E. Shirokov
Steklov Mathematical Institute, RAS, Moscow
Received January 12, 2006; in ﬁnal form, August 8, 2006
Abstract—Special sets of states, called optimal, which are related to the Holevo capacity
and to the minimal output entropy of a quantum channel, are considered. By methods of
convex analysis and operator theory, structural properties of optimal sets and conditions of
their coincidence are explored for an arbitrary channel. It is shown that strong additivity of
the Holevo capacity for two given channels provides projective relations between optimal sets
for the tensor product of these channels and optimal sets for the individual channels.
One of the main notions of quantum information theory is that of a quantum channel, trans-
forming states of a quantum system into states of another quantum system. The minimal output
entropy and the Holevo capacity
are important characteristics of a quantum channel . Recently,
a lot of papers devoted to the study of these characteristics were published [3–5]. In the present
paper, we analyze the structure of two special sets of states of an input quantum system related to
the above characteristics of a channel, pointing the main attention to the case of a tensor product
The paper is organized as follows. In Section 2 it is shown that the minimal output entropy and
the Holevo capacity can be expressed via the Fenchel transform of the output entropy of a channel
(Lemma 1). This observation and properties of optimal ensembles  provide a characterization
of the average state of an optimal ensemble for an arbitrary quantum channel in terms of convex
duality (Proposition 1).
In Section 3, for an arbitrary quantum channel we consider two optimal sets of states, related
to the Holevo capacity and to the minimal output entropy of this channel, respectively. Some
properties of these sets and a necessary and suﬃcient condition for their coincidence are considered
(Proposition 2, Theorem 1, and Corollary 1). By using this condition, a characterization of a
channel for which the optimal ensemble consists of optimal states is obtained (Corollary 2).
One of the most interesting open problems of quantum information theory is the additivity
conjecture for the Holevo capacity for all quantum channels . Recently it was proved  that
this conjecture is equivalent to the additivity conjecture for the minimal output entropy for all
channels. In this paper, relations between the assumed additivity of the Holevo capacity and
of the minimal output entropy for two channels, as well as the structure of optimal sets for the
Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00164a, and the
program “Modern Problems of Theoretical Mathematics” of the Russian Academy of Sciences.
This characteristic, also called the χ-capacity, is closely related to the capacity in the case of classical
information transmitted over a quantum channel [1, 2].