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arXiv:2302.04809 · Sbornik Mathematics

Lower semicontinuity of relative entropy disturbance and its consequences

Максим Евгеньевич Широков

Published Jan 1, 2024• 0 citations
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Abstract

It is proved that the decrease of quantum relative entropy under the action of a quantum operation is a lower semicontinuous function of the pair of its arguments. This property implies, in particular, that the local discontinuity jumps of the quantum relative entropy do not increase under the action of quantum operations. It also implies the lower semicontinuity of the modulus of the joint convexity of quantum relative entropy (as a function of ensembles of quantum states). Various corollaries and applications of these results are considered. Bibliography: 42 titles.

DOI
10.4213/sm10107e

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It is proved that the decrease of quantum relative entropy under the action of a quantum operation is a lower semicontinuous function of the pair of its arguments. This property implies, in particular, that the local discontinuity jumps of the quantum relative entropy do not increase under the action of quantum operations. It also implies the lower semicontinuity of the modulus of the joint convexity of quantum relative entropy (as a function of ensembles of quantum states). Various corollaries and applications of these results are considered. Bibliography: 42 titles.
We prove that for any infinite-dimensional quantum channel the entropic disturbance (defined as difference between the χχχ-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-χ\textrm{-}χ-quantity and the existence of χχχ-optimal ensemble for any quantum channel under the energy-type input constraint.
Abstract It is shown that the recently established lower semicontinuity of the quantum conditional mutual information implies (in fact, is equivalent to) the lower semicontinuity of the loss of the quantum (conditional) mutual information under local channels considered as a function on the Cartesian product of the set of all states of a composite system and the sets of all local channels (equipped with the strong convergence topology). Some applications of this property are considered. New continuity conditions for the quantum mutual information and for the squashed entanglement in both bipartite and multipartite infinite-dimensional systems are obtained. It is proved, in particular, that the multipartite squashed entanglement of any countably indecomposable separable state with finite marginal entropies is equal to zero. Special continuity properties of the information gain of a quantum measurement with and without quantum side information are established that can be treated as robustness (stability) of these quantities with respect to perturbation of the measurement and the measured state.
Special approximation technique for analysis of different characteristics of states of multipartite infinite-dimensional quantum systems is proposed and applied to study of the relative entropy of entanglement and its regularisation. We prove several results about analytical properties of the multipartite relative entropy of entanglement and its regularization (the lower semicontinuity on wide class of states, the uniform continuity under the energy constraints, etc.). We establish a finite-dimensional approximation property for the relative entropy of entanglement and its regularization that allows to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.
Quantitative analysis of discontinuity of information characteristics of quantum states and channels is presented. Estimates for discontinuity jump (loss) of the von Neumann entropy for a given converging sequence of states are obtained. It is shown, in particular, that for any sequence the loss of entropy is upper bounded by the loss of mean energy (with the coefficient characterizing the Hamiltonian of a system). Then we prove that discontinuity jumps of basic measures of classical and quantum correlations in composite quantum systems are upper bounded by the loss of one of the marginal entropies (with a corresponding coefficient). Quantitative discontinuity analysis of the output entropy of a quantum operation and of basic information characteristics of a quantum channel considered as functions of a pair (channel, input state) is presented.