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The continuity condition for the von Neumann entropy based on the special approximation

Максим Евгеньевич Широков

Published Apr 13, 2009• 2 citations
OpenAlex

Abstract

The universal method of proving continuity of the von Neumann entropy on subsets of positive trace-class operators are considered. It makes possible to re-derive the known conditions of continuity of the entropy in the more general forms and to obtain the several new conditions. This method is based on the special approximation of the von Neumann entropy by the increasing sequence of concave continuous unitary invariant functions. Existence of this approximation is a corollary of the general property of the set of quantum states as a convex topological space called the strong stability property and considered in the first part of the paper.

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Continuity condition for concave functions on convex μμμ-compact sets and its applications in quantum physics
A method of proving local continuity of concave functions on convex set possessing the μμμ-compactness property is presented. This method is based on a special approximation of these functions. The class of μμμ-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which particular results well known for compact sets can be generalized. Applications of the obtained continuity conditions to analysis of different entropic characteristics of quantum systems and channels are considered.
It is well known that the von Neumann entropy is continuous on a subset of quantum states with bounded energy provided the Hamiltonian HHH of the system satisfies the condition $\Tr\exp(-cH)0$. In this note we consider the following conjecture: every closed convex subset of quantum states, on which the von Neumann entropy is continuous, consists of states with bounded energy with respect to a particular Hamiltonian HHH satisfying the above condition. It is shown that the classical analog of this conjecture is valid (i.e. it is valid for the Shannon entropy). It is also shown that this conjecture holds for some types of subsets consisting of non-commuting states, but its validity for all subsets of quantum states remains an open question.
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We derive a measurement-independent asymptotic continuity bound on the observational entropy for general positive operator valued measures measurements, making essential use of its property of bounded concavity. The same insight is used to obtain continuity bounds for other entropic quantities, including the measured relative entropy distance to a convex set of states under a general set of measurements. As a special case, we define and study conditional observational entropy, which is an observational entropy in one (measured) subsystem conditioned on the quantum state in another (unmeasured) subsystem. We also study continuity of relative entropy with respect to a jointly applied channel, finding that observational entropy is uniformly continuous as a function of the measurement. But we show by means of an example that this continuity under measurements cannot have the form of a concrete asymptotic bound.