On continuity properties of the entropy of an observable

Marian Grabowski

Pierre Gaspard

Gisela A. Lassner, Gisela A. Lassner

Joseph Schindler, Andreas Winter

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.

Colette Padet

In our previous papers, we proposed an analysis of the relations between information theory and observation theory. Referring to the parameters of the observation and using Shannon's formulation, we defined and calculated the information associated with a result of observation and the entropy of a random variable for a process of observation. This paper presents an extension of this theory for other definitions of entropy (Onicescu, hyperbolic). The method allows in particular to define Onicescu and hyperbolic entropies of continuous random variables. We examine the main properties of these new functions and propose a comparative analysis.

ZhiHua Chen, Zhihao Ma, Ismail Nikoufar +1 more