arXiv:1608.02203 · Izvestiya Mathematics

On lower semicontinuity of the entropic disturbance and its applications in quantum information theory

Максим Евгеньевич Широков, A S Holevo

Published Oct 31, 2017• 5 citations

Abstract

We prove that for any infinite-dimensional quantum channel the entropic disturbance (defined as difference between the $χ$-quantity of a generalized ensemble and that of the image of the ensemble under the channel) is lower semicontinuous on the natural set of its definition. We establish a number of useful corollaries of this property, in particular, we prove the continuity of the output $χ\textrm{-}$quantity and the existence of $χ$-optimal ensemble for any quantum channel under the energy-type input constraint.

DOI: 10.1070/im8609

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arXiv:1608.02203 · Izvestiya Mathematics

On lower semicontinuity of the entropic disturbance and its applications in quantum information theory

Максим Евгеньевич Широков, A S Holevo

Published Oct 31, 2017• 5 citations

arXiv HTML PDF

Abstract

We prove that for any infinite-dimensional quantum channel the entropic disturbance (defined as difference between the

χ

-quantity of a generalized ensemble and that of the image of the ensemble under the channel) is lower semicontinuous on the natural set of its definition. We establish a number of useful corollaries of this property, in particular, we prove the continuity of the output

χ\textrm{-}

quantity and the existence of

χ

-optimal ensemble for any quantum channel under the energy-type input constraint.

DOI: 10.1070/im8609

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Theory of Probability and Its ApplicationsJan 1, 200815 citations

We prove that for any infinite-dimensional quantum channel the entropic disturbance (defined as difference between the

χ

χ\textrm{-}

quantity and the existence of

χ

-optimal ensemble for any quantum channel under the energy-type input constraint.

We show that a sequence

\{\Phi_n\}

of quantum channels strongly converges to a quantum channel

\Phi_0

if and only if there exist a common environment for all the channels and a corresponding sequence

\{V_n\}

of Stinespring isometries strongly converging to a Stinespring isometry

V_0

of the channel

\Phi_0

. We also give a quantitative description of the above characterization of the strong convergence in terms of the appropriate metrics on the sets of quantum channels and Stinespring isometries. As a result, the uniform selective continuity of the complementary operation with respect to the strong convergence is established. We show discontinuity of the unitary dilation by constructing a strongly converging sequence of channels which can not be represented as a reduction of a strongly converging sequence of unitary channels. The Stinespring representation of strongly converging sequences of quantum channels allows to prove the lower semicontinuity of the entropic disturbance as a function of a pair (channel, input ensemble). Some corollaries of this property are considered.

\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.