ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 2, pp. 73–90.
Pleiades Publishing, Inc., 2008.
Original Russian Text
M.E. Shirokov, A.S. Holevo, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 2, pp. 3–22.
On Approximation of Inﬁnite-Dimensional
M. E. Shirokov and A. S. Holevo
Steklov Mathematical Institute, RAS, Moscow
Received December 11, 2007
Abstract—We develop an approximation approach to inﬁnite-dimensional quantum channels
based on a detailed investigation of continuity properties of entropic characteristics of quantum
channels and operations (trace-nonincreasing completely positive maps) as functions of a pair
“channel, input state.” Obtained results are then applied to the problems of continuity of the
χ-capacity as a function of a channel, strong additivity of the χ-capacity for inﬁnite-dimensional
channels, and approximating representation for the convex closure of the output entropy of an
arbitrary quantum channel.
Though major attention in quantum information theory was so far paid to ﬁnite-dimensional
systems and channels, there is an increasing interest in inﬁnite-dimensional generalizations (see
[1–7] and references therein). An essential feature of inﬁnite-dimensional channels is discontinuity
and unboundedness of main entropic characteristics, which makes a straightforward generalization
of results obtained in ﬁnite dimensions impossible. A natural way to study inﬁnite-dimensional
quantum channels is to approximate them in an appropriate topology by channels with continuous
characteristics (for example, channels with ﬁnite-dimensional output spaces). This approach was
(implicitly) used in  to derive strong additivity of the Holevo capacity (χ-capacity in what follows)
for some classes of inﬁnite-dimensional channels from the corresponding ﬁnite-dimensional results
and to prove that validity of the additivity conjecture in ﬁnite dimensions implies strong additivity
of the χ-capacity for all inﬁnite-dimensional channels.
In the present paper we develop an approximation approach to inﬁnite-dimensional quan-
tum channels based on a detailed investigation of continuity properties of entropic characteris-
tics of quantum channels related to the classical capacity as functions of a pair “channel, in-
put state”. It appears that often it is convenient to approximate a channel by operations, i.e.,
trace-nonincreasing completely positive maps, rather than by channels (from the point of view of
noncommutative probability, an operation is a sub-Markov map, while a channel is a Markov map).
Thus, we have to extend deﬁnitions of entropic characteristics to operations and study continuity
properties of these characteristics on the extended domain.
The paper is organized as follows. Section 2 presents basic notions and some results of previous
works used in this paper. In Section 3 we consider the topology of strong convergence on the set of
all quantum operations, which appears to be a proper topology for approximation purposes. It is
Supported in part by the program “Modern Problems of Theoretical Mathematics” of the Russian Academy
of Sciences, the Russian Foundation for Basic Research, project no. 06-01-00164-a, and NSH, grant