Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds

TL;DR

This paper explores the mathematical and information-theoretic properties of observational entropy in quantum systems, establishing bounds and identities that relate measurement outcomes, coarse-grained states, and entropy differences.

Contribution

It introduces new bounds and identities for observational entropy using strengthened monotonicity of quantum relative entropy, emphasizing the role of coarse-grained states derived from measurement statistics.

Findings

Derived bounds on observational entropy applicable in general settings

Identified relationships between coarse-grained states and true quantum states

Established bounds on the difference between observational and von Neumann entropies

Abstract

Observational entropy provides a general notion of quantum entropy that appropriately interpolates between Boltzmann's and Gibbs' entropies, and has recently been argued to provide a useful measure of out-of-equilibrium thermodynamic entropy. Here we study the mathematical properties of observational entropy from an information-theoretic viewpoint, making use of recently strengthened forms of the monotonicity property of quantum relative entropy. We present new bounds on observational entropy applying in general, as well as bounds and identities related to sequential and post-processed measurements. A central role in this work is played by what we call the ``coarse-grained'' state, which emerges from the measurement's statistics by Bayesian retrodiction, without presuming any knowledge about the ``true'' underlying state being measured. The degree of distinguishability between such a coarse-grained state and the true (but generally unobservable) one is shown to provide upper and lower bounds on the difference between observational and von Neumann entropies.