Observational entropy with general quantum priors

TL;DR

This paper explores the concept of observational entropy in quantum systems, offering new generalizations that replace the uniform prior with arbitrary quantum states, applicable to infinite-dimensional or energy-constrained systems.

Contribution

It introduces three new generalizations of observational entropy using non-uniform priors, unifying different interpretations and extending applicability to complex quantum systems.

Findings

Proposes three candidate generalizations of observational entropy.

Shows one generalization unifies measurement deficiency and retrodiction perspectives.

Extends the concept to infinite-dimensional and energy-constrained quantum systems.

Abstract

Observational entropy captures both the intrinsic uncertainty of a thermodynamic state and the lack of knowledge due to coarse-graining. We demonstrate two interpretations of observational entropy, one as the statistical deficiency resulting from a measurement, the other as the difficulty of inferring the input state from the measurement statistics by quantum Bayesian retrodiction. These interpretations show that the observational entropy implicitly includes a uniform reference prior. Since the uniform prior cannot be used when the system is infinite-dimensional or otherwise energy-constrained, we propose generalizations by replacing the uniform prior with arbitrary quantum states that may not even commute with the state of the system. We propose three candidates for this generalization, discuss their properties, and show that one of them gives a unified expression that relates both interpretations.