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arXiv:quant-ph/0411091 · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

On entropic quantities related to the classical capacity of infinite dimensional quantum channels

Максим Евгеньевич Широков

Published Jun 1, 2005• 11 citations
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Abstract

The χ-function (the Holevo capacity of constrained channel) and the convex closure of the output entropy for arbitrary infinite dimensional channel are considered. The properties of the χ-function and of the convex closure of the output entropy are explored. The applications of the obtained results to the additivity problem and to the theory of entanglement are presented.

DOI
10.1117/12.620512

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The χ-function (the Holevo capacity of constrained channel) and the convex closure of the output entropy for arbitrary infinite dimensional channel are considered. The properties of the χ-function and of the convex closure of the output entropy are explored. The applications of the obtained results to the additivity problem and to the theory of entanglement are presented.
This paper is devoted to further study of the Holevo capacity of infinite-dimensional quantum channels. The existence of a unique optimal average state for a quantum channel constrained by an arbitrary convex set of states is shown. The minimax expression for the Holevo capacity of a constrained channel is obtained. The χ\chiχ-function and the convex closure of the output entropy of an infinite-dimensional quantum channel are considered. It is shown that the χ\chiχ-function of an arbitrary channel is lower semicontinuous on the set of all states and has continuous restrictions to subsets of states with continuous output entropy. The explicit expression for the convex closure of the output entropy of an infinite-dimensional quantum channel is obtained and its properties are explored. It is shown that the convex closure of the output entropy coincides with the convex hull of the output entropy on the set of states with finite output entropy and, similarly to the χ\chiχ-function, it has continuous restrictions to subsets of states with continuous output entropy. The applications of the obtained results to the theory of entanglement are considered. The properties of the convex closure of the output entropy make it possible to generalize some results related to the additivity problem to the infinite-dimensional case.
We develop an approximation approach to infinite dimensional quantum channels based on detailed investigation of the continuity properties of entropic characteristics of quantum channels and operations (trace-nonincreasing completely positive maps) as functions of a pair ``channel, input state''. The obtained results are then applied to the following problems: continuity of the χχχ-capacity as function of a channel; strong additivity of the χχχ-capacity for infinite dimensional channels; the analytical expression for the convex closure of the output entropy of arbitrary quantum channel.
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.
The notion of the Holevo capacity for arbitrarily constrained infinite dimensional quantum channels is introduced. It is shown that despite nonexistence of an optimal ensemble in this case it is possible to define the notion of the output optimal average state for such a channel. The characterization of the output optimal average state and a minimax expression for the Holevo capacity are obtained. This makes it possible to prove equivalence of several additivity properties for infinite dimensional quantum channels. The notion of the χ\chiχ-function for an infinite dimensional channel is considered, its strong concavity and lower semicontinuity are shown. The problem of continuity of the Holevo capacity is also discussed. It is shown that the Holevo capacity is continuous function of a channel in the finite dimensional case while in general it is only lower semicontinuous. This conclusion is confirmed by the example. The main result of this note is the statement that additivity of the Holevo capacity for all finite dimensional channels implies additivity of the Holevo capacity for all infinite dimensional channels with arbitrary constraints. The subadditivity of the χ\chiχ-function for two infinite dimensional channels with one of them noiseless or entanglement breaking is also proved.
The continuity properties of the convex closure of the output entropy of infinite dimensional channels and their applications to the additivity problem are considered. The main result of this paper is the statement that the superadditivity of the convex closure of the output entropy for all finite dimensional channels implies the superadditivity of the convex closure of the output entropy for all infinite dimensional channels, which provides the analogous statements for the strong superadditivity of the EoF and for the additivity of the minimal output entropy. The above result also provides infinite dimensional generalization of Shor's theorem stated equivalence of different additivity properties. The superadditivity of the convex closure of the output entropy (and hence the additivity of the minimal output entropy) for two infinite dimensional channels with one of them a direct sum of noiseless and entanglement-breaking channels are derived from the corresponding finite dimensional results. In the context of the additivity problem some observations concerning complementary infinite dimensional channels are considered.
A coding theorem for entanglement-assisted communication via an infinite-dimensional 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.