ISSN 0032-9460, Problems of Information Transmission, 2013, Vol. 49, No. 1, pp. 15–31.
Pleiades Publishing, Inc., 2013.
Original Russian Text
A.S. Holevo, M.E. Shirokov, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 1, pp. 19–36.
On Classical Capacities of Inﬁnite-Dimensional
A. S. Holevo and M. E. Shirokov
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow
Received October 22, 2012
Abstract—A coding theorem for entanglement-assisted communication via an inﬁnite-dimen-
sional quantum channel with linear constraints is extended to a natural degree of generality.
Relations between the entanglement-assisted classical capacity and χ-capacity of constrained
channels are obtained, and conditions for their coincidence are given. Suﬃcient conditions for
continuity of the entanglement-assisted classical capacity as a function of a channel are obtained.
Some applications of the obtained results to analysis of Gaussian channels are considered.
A general (continuous) version of the fundamental relation between coherent information and
the measure of privacy of classical information transmission via an inﬁnite-dimensional quantum
channel is proved.
A central role in quantum information theory is played by the notion of a quantum channel,
a noncommutative analog of a transition probability matrix in classical theory. Informational prop-
erties of a quantum channel are characterized by a number of diﬀerent capacities depending on the
type of transmitted information, additional resources used to increase the rate of transmission,
security requirements, etc.; see, e.g., . One of the most important of these quantities is the
entanglement-assisted classical capacity, which characterizes the ultimate rate of classical informa-
tion transmission assuming that the transmitter and receiver may use a common entangled state.
By deﬁnition, this capacity is greater than or equal to the classical (unassisted) capacity of the
channel. The Bennett–Shor–Smolin–Thapliyal (BSST) theorem  gives an explicit expression for
the entanglement-assisted capacity of a ﬁnite-dimensional unconstrained channel, showing that this
capacity is equal to the maximum of quantum mutual information.
When applying the protocol of entanglement-assisted communication to inﬁnite-dimensional
channels, one has to impose certain constraints on input states. A typical physically motivated
constraint is bounded energy of states used for encoding. This constraint is determined by the
Tr ρF ≤ E, E > 0, (1)
where F is a positive self-adjoint operator, the Hamiltonian of the input quantum system. An oper-
ational deﬁnition of the entanglement-assisted classical capacity of an inﬁnite-dimensional quantum
channel with linear constraint (1) is given in , where a generalization of the BSST theorem is
proved under special restrictions on the channel and on the constraint operator. Recent advances
in the study of entropy characteristics of inﬁnite-dimensional quantum channels (in particular,
Supported in part by the Russian Foundation for Basic Research, project nos. 12-01-00319-a and 13-01-