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The Holevo capacity of infinite dimensional channels

Максим Евгеньевич Широков

Published Aug 2, 2004• 5 citations
OpenAlex

Abstract

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 $\chi$-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 $\chi$-function for two infinite dimensional channels with one of them noiseless or entanglement breaking is also proved.

Related papers

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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 $\chi$-function and the convex closure of the output entropy of an infinite-dimensional quantum channel are considered. It is shown that the $\chi$-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 $\chi$-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.

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

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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.

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Continuous ensembles and the $χ$-capacity of infinite-dimensional channels

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The paper is devoted to systematic study of the $χ$-capacity (underlying the classical capacity) of infinite dimensional quantum channels. An essential feature of this case is the natural appearance of the input constraints and infinite, in general, ``continuous'' state ensembles, defined as probability measures on the set of all quantum states. By using compactness criteria from probability theory and operator theory it is shown that the set of all generalized ensembles with the average (barycenter) in a compact set of states is itself a compact subset of the set of all probability measures. With this in hand we give a sufficient condition for the existence of an optimal generalized ensemble for a constrained quantum channel. This condition can be verified in the case of Bosonic Gaussian channels with constrained mean energy. The importance of the above condition is shown by considering example of a constrained channel with no optimal generalized ensemble. In the case of convex constraints a characterization of the optimal generalized ensemble is obtained extending the `` maximal distance'' property.

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Continuous Ensembles and the Capacity of Infinite-Dimensional Quantum Channels

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This paper is devoted to the study of $\chi$-capacity, closely related to the classical capacity of infinite-dimensional quantum channels. For such channels generalized ensembles are defined as probability measures on the set of all quantum states. We establish the compactness of the set of generalized ensembles with averages in an arbitrary compact subset of states. This result enables us to obtain a sufficient condition for the existence of the optimal generalized ensemble for an infinite-dimensional channel with input constraint. This condition is shown to be fulfilled for Bosonic Gaussian channels with constrained mean energy. In the case of convex constraints, a characterization of the optimal generalized ensemble extending the "maximal distance property" is obtained.

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The Convex Closure of the Output Entropy of Infinite Dimensional Channels and the Additivity Problem

Максим Евгеньевич Широков

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.

arXiv:quant-ph/0608090•Aug 10, 2006•3 citations

On the additivity conjecture for channels with arbitrary constraints

Максим Евгеньевич Широков

Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property of an optimal ensemble for such channel is derived, generalizing the maximal distance property of Schumacher and Westmoreland (proposition 1). It is shown that the additivity conjecture for two channels with single linear constraints is equivalent to the similar conjecture for two arbitrarily constrained channels and, hence, to an interesting subadditivity property of the $χ$-function for the tensor product of these channels (theorem 1). We also propose an alternative way of proving that the additivity conjecture for any two unconstrained channels implies strong superadditivity of the entanglement of formation. The arguments from the convex analysis provide another characterization of channels for which subadditivity of the $χ$-function holds (theorem 3). This characterization and some modification of Shor's channel extension provide a simple way of proving that global additivity of the minimum output entropy for unconstrained channels implies global subadditivity of the $χ$-function and strong superadditivity of the entanglement of formation.

arXiv:quant-ph/0308168•Aug 29, 2003•3 citations
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.
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.
Continuous ensembles and the χχχ-capacity of infinite-dimensional channels
The paper is devoted to systematic study of the χχχ-capacity (underlying the classical capacity) of infinite dimensional quantum channels. An essential feature of this case is the natural appearance of the input constraints and infinite, in general, ``continuous'' state ensembles, defined as probability measures on the set of all quantum states. By using compactness criteria from probability theory and operator theory it is shown that the set of all generalized ensembles with the average (barycenter) in a compact set of states is itself a compact subset of the set of all probability measures. With this in hand we give a sufficient condition for the existence of an optimal generalized ensemble for a constrained quantum channel. This condition can be verified in the case of Bosonic Gaussian channels with constrained mean energy. The importance of the above condition is shown by considering example of a constrained channel with no optimal generalized ensemble. In the case of convex constraints a characterization of the optimal generalized ensemble is obtained extending the `` maximal distance'' property.
This paper is devoted to the study of χ\chiχ-capacity, closely related to the classical capacity of infinite-dimensional quantum channels. For such channels generalized ensembles are defined as probability measures on the set of all quantum states. We establish the compactness of the set of generalized ensembles with averages in an arbitrary compact subset of states. This result enables us to obtain a sufficient condition for the existence of the optimal generalized ensemble for an infinite-dimensional channel with input constraint. This condition is shown to be fulfilled for Bosonic Gaussian channels with constrained mean energy. In the case of convex constraints, a characterization of the optimal generalized ensemble extending the "maximal distance property" is obtained.
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.
Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property of an optimal ensemble for such channel is derived, generalizing the maximal distance property of Schumacher and Westmoreland (proposition 1). It is shown that the additivity conjecture for two channels with single linear constraints is equivalent to the similar conjecture for two arbitrarily constrained channels and, hence, to an interesting subadditivity property of the χχχ-function for the tensor product of these channels (theorem 1). We also propose an alternative way of proving that the additivity conjecture for any two unconstrained channels implies strong superadditivity of the entanglement of formation. The arguments from the convex analysis provide another characterization of channels for which subadditivity of the χχχ-function holds (theorem 3). This characterization and some modification of Shor's channel extension provide a simple way of proving that global additivity of the minimum output entropy for unconstrained channels implies global subadditivity of the χχχ-function and strong superadditivity of the entanglement of formation.