Entropy, majorization and thermodynamics in general probabilistic theories

TL;DR

This paper develops a framework for thermodynamics in general probabilistic theories, defining spectral entropy and conditions under which measurement outcomes are majorized, extending quantum thermodynamic concepts to broader theories.

Contribution

It introduces spectral entropy in GPTs satisfying spectrality, projectivity, and symmetry, and extends quantum thermodynamic entropy concepts to these generalized systems.

Findings

Spectral entropy equals measurement entropy under certain conditions.

Projectivity and symmetry lead to a geometric property called perfection.

Extension of von Neumann's thermodynamic entropy argument to GPTs.

Abstract

In this note we lay some groundwork for the resource theory of thermodynamics in general probabilistic theories (GPTs). We consider theories satisfying a purely convex abstraction of the spectral decomposition of density matrices: that every state has a decomposition, with unique probabilities, into perfectly distinguishable pure states. The spectral entropy, and analogues using other Schur-concave functions, can be defined as the entropy of these probabilities. We describe additional conditions under which the outcome probabilities of a fine-grained measurement are majorized by those for a spectral measurement, and therefore the "spectral entropy" is the measurement entropy (and therefore concave). These conditions are (1) projectivity, which abstracts aspects of the Lueders-von Neumann projection postulate in quantum theory, in particular that every face of the state space is the positive part of the image of a certain kind of projection operator called a filter; and (2) symmetry of transition probabilities. The conjunction of these, as shown earlier by Araki, is equivalent to a strong geometric property of the unnormalized state cone known as perfection: that there is an inner product according to which every face of the cone, including the cone itself, is self-dual. Using some assumptions about the thermodynamic cost of certain processes that are partially motivated by our postulates, especially projectivity, we extend von Neumann's argument that the thermodynamic entropy of a quantum system is its spectral entropy to generalized probabilistic systems satisfying spectrality.