arXiv subjects

M. E. Shirokov

Publications and source records attributed to M. E. Shirokov.

80 records · Page 5Linked to original sources

arXiv2005

Entropic characteristics of subsets of states

This paper is devoted to systematic study of properties of the quantum entropy and of the Holevo capacity considered as a function of a set of quantum states. The properties of restriction of the quantum entropy to arbitrary set of states are discussed. For some types of sets necessary and sufficient conditions of boundedness and of continuity of restriction of the quantum entropy as well as necessary and sufficient conditions of existence of the Gibbs state are obtained. The Holevo capacity of an arbitrary set of states and its general properties as a function of a set are considered. The notion of the optimal average state of an arbitrary set of states with finite Holevo capacity is introduced. The condition of existence of optimal measure for a set of states is obtained. The general results concerning the quantum entropy and the Holevo capacity are illustrated by considering the several examples of sets of states.

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arXiv2005

On the structure of optimal sets for tensor product channel

For a given quantum channel we consider two optimal sets of states, related to the Holevo capacity and to the minimal output entropy of this channel. Some properties of these sets as well as the necessary and sufficient condition for their coincidence are obtained. The relations between additivity properties for two quantum channels and the structure of the optimal sets for tensor product of these channels are considered. It turns out that these additivity properties are connected with the special hereditary property of the optimal sets for tensor product channel. We explore the structural properties of these optimal sets under two different assumptions. The first assumption means that for tensor product of two channels with arbitrary constraints there exists optimal ensemble with the product state average. It turns out that exactly this assumption guarantees hereditary property of the both optimal sets for tensor product channel and provides some observations concerning the additivity problems. The second assumption means additivity of the Holevo capacity for two channels with arbitrary constraints. It turns out that exactly this assumption guarantees strong hereditary property of the both optimal sets for tensor product channel and provides the natural "projective" relations between these sets and the optimal sets for single channels.

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arXiv2005

Separability and Entanglement-Breaking in Infinite Dimensions

In this paper we give a general integral representation for separable states in the tensor product of infinite dimensional Hilbert spaces and provide the first example of separable states that are not countably decomposable. We also prove the structure theorem for the quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of M. Horodecki, Ruskai and Shor. In the finite dimensional case such channels can be characterized as having the Kraus representation with operators of rank 1. The above example implies existence of infinite-dimensional entanglement-breaking channels having no such representation.

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arXiv2005

The Holevo capacity of infinite dimensional channels and the additivity problem

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

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arXiv2004

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

In this paper we consider the $χ$-function (the Holevo capacity of constrained channel) and the convex closure of the output entropy for arbitrary infinite dimensional channel. It is shown that the $χ$-function of an arbitrary channel is a concave lower semicontinuous function on the whole state space, having continuous restriction to any set of continuity of the output entropy. The explicit representation for the convex closure of the output entropy 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 convex set of states with finite output entropy. Similarly to the case of the $χ$-function, it is proved that the convex closure of the output entropy has continuous restriction to any set of continuity of output entropy. Some applications of these results to the theory of entanglement are discussed. The obtained properties of the convex closure of the output entropy make it possible to generalize to the infinite dimensional case the convex duality approach to the additivity problem.

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arXiv2004

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

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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arXiv2004

On Shor's channel extension and constrained channels

In this paper we give several equivalent formulations of the additivity conjecture for constrained channels, which formally is substantially stronger than the unconstrained additivity. To this end a characteristic property of the optimal ensemble for such a channel is derived, generalizing the maximal distance property. It is shown that the additivity conjecture for constrained channels holds true for certain nontrivial classes of channels. Recently P. Shor showed that conjectured additivity properties for several quantum information quantities are in fact equivalent. After giving an algebraic formulation for the Shor's channel extension, its main asymptotic property is proved. It is then used to show that additivity for two constrained channels can be reduced to the same problem for unconstrained channels, and hence, "global" additivity for channels with arbitrary constraints is equivalent to additivity without constraints.

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arXiv2003

On the additivity conjecture for channels with arbitrary constraints

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.

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