ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 3, pp. 171–184.
Pleiades Publishing, Inc., 2008.
Original Russian Text
A.S. Holevo, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 3, pp. 3–18.
in Inﬁnite Dimensions
A. S. Holevo
Steklov Mathematical Institute, RAS, Moscow
Received February 22, 2008; in ﬁnal form, April 25, 2008
Abstract—In the ﬁrst part of the paper we give a representation for entanglement-breaking
channels in separable Hilbert space that generalizes the “Kraus decomposition with rank-one
operators” and use it to describe complementary channels. We also note that coherent informa-
tion for antidegradable channel is always nonpositive. In the second part, we give necessary and
suﬃcient condition for entanglement breaking for the general quantum Gaussian channel. Ap-
plication of this condition to one-mode channels provides several new cases where the additivity
conjecture holds in the strongest form.
One of the key notions of quantum information theory is entanglement, which is a speciﬁc kind of
correlation, absent in classical systems. An important and well-understood class of quantum com-
munication channels in ﬁnite dimensions is the class of entanglement-breaking channels (in what
follows, EB channels), studied in detail in , where it was shown that the property of entangle-
ment breaking is equivalent to the structural property of a channel to have intermediate classical
stage: any such channel can be represented as a quantum measurement of an input state followed
by preparing an output state depending on the outcome of the measurement. Thus, entanglement-
breaking channels precisely coincide with a class of channels introduced in . Another characteris-
tic feature of entanglement-breaking channels found in  is the existence of the Kraus decomposi-
tion with rank-one operators. The present paper is devoted to the study of entanglement-breaking
channels in inﬁnite dimensions and has two self-consistent parts. In part I (Sections 2 and 3) we
give a proper generalization of the Kraus decomposition with rank-one operators in the case of a
separable Hilbert space and use it to describe complementary channels. Part II (Sections 4–6) is
devoted to Gaussian entanglement-breaking channels. We give necessary and suﬃcient condition
for entanglement breaking for the general quantum Gaussian channel. Application of this condition
to one-mode channels provides several new cases where the famous additivity conjecture holds in
the strongest form.
In  we gave the general integral representation for separable (unentangled) states in a tensor
product of inﬁnite-dimensional Hilbert spaces and proved a structure theorem, which gives a con-
tinual version of channels introduced in . It was also shown that a straightforward extension of
the Kraus decomposition with rank one operators is not valid in the inﬁnite-dimensional case.
In what follows, H denotes a separable Hilbert space, T(H) is the Banach space of trace-class
operators in H,andS(H) is the convex subset of all density operators ρ in H.Wealsocall
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.