What is Entropy? A new perspective from games of chance

TL;DR

This paper introduces a new perspective on entropy by linking it to games of chance, constructing families of games that induce majorization pre-orders, and providing operational interpretations relevant to dynamical resource theories.

Contribution

It develops a systematic approach to defining entropy through games of chance, connecting majorization concepts with operational interpretations and dynamical resource theories.

Findings

Constructed families of games inducing majorization pre-orders

Provided operational interpretations for these pre-orders

Identified the only asymptotically continuous classical dynamic entropy

Abstract

Given entropy's central role in multiple areas of physics and science, one important task is to develop a systematic and unifying approach to defining entropy. Games of chance become a natural candidate for characterising the uncertainty of a physical system, as a system's performance in gambling games depends solely on the uncertainty of its output. In this work, we construct families of games which induce pre-orders corresponding to majorization, conditional majorization, and channel majorization. Finally, we provide operational interpretations for all pre-orders, show the relevance of these results to dynamical resource theories, and find the only asymptotically continuous classical dynamic entropy.