Continuity bounds on observational entropy and measured relative entropies

TL;DR

This paper establishes measurement-independent continuity bounds for observational entropy and related quantities, demonstrating their stability under measurement changes and exploring their properties in quantum systems.

Contribution

It introduces a novel asymptotic continuity bound for observational entropy that is measurement-independent and extends the analysis to other entropic measures and conditional entropy.

Findings

Derived a measurement-independent asymptotic continuity bound for observational entropy.

Established continuity bounds for measured relative entropy distance to convex sets of states.

Showed observational entropy's uniform continuity as a function of measurement, with limitations on asymptotic bounds.

Abstract

We derive a measurement-independent asymptotic continuity bound on the observational entropy for general POVM 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 a 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.