Resource Theories as Quantale Modules

TL;DR

This paper develops an abstract mathematical framework using quantale modules to unify and generalize resource theories across different scientific contexts, enabling the transfer and extension of resource measures.

Contribution

It introduces two variants of an abstract framework for resource theories using quantale modules, facilitating the translation and generalization of resource measures across contexts.

Findings

Framework allows construction of resource cost and yield measures.

Enables translation of resource measures between theories.

Applied to a resource theory of distinguishability.

Abstract

We aim to counter the tendency for specialization in science by advancing a language that can facilitate the translation of ideas and methods between disparate contexts. The focus is on questions of "resource-theoretic nature". In a resource theory, one identifies resources and allowed manipulations that can be used to transform them. Some of the main questions are: How to optimize resources? What are the trade-offs between them? Can a given resource be converted to another one via the allowed manipulations? Because of their ubiquity, methods used to answer them in one context can be used to tackle corresponding questions in new contexts. The translation occurs in two stages. Firstly, methods are generalized to the abstract language. Then, one can determine whether potentially novel contexts can accommodate them. We focus on the first stage, by introducing two variants of an abstract framework in which existing and yet unidentified resource theories can be represented. Using these, the task of generalizing concrete methods is tackled in chapter 4 by studying the ways in which meaningful measures of resources may be constructed. One construction expresses a notion of cost (or yield) of a resource. Among other applications, it may be used to extend measures from a subset of resources to a larger domain. Another construction allows the translation of resource measures from one resource theory to another. Special cases include resource robustness and weight measures, as well as relative entropy based measures quantifying minimal distinguishability from freely available resources. We instantiate some of these ideas in a resource theory of distinguishability in chapter 5. It describes the utility of systems with probabilistic behavior for the task of distinguishing between hypotheses, which said behavior may depend on.