A mathematical theory of resources

TL;DR

This paper develops a unified mathematical framework for resource theories across various scientific fields, enabling analysis of resource convertibility, quantification, and the role of catalysts.

Contribution

It introduces a general definition of resource theories based on process theories with free processes, unifying existing theories and identifying universal principles.

Findings

Framework encompasses quantum, thermodynamic, and communication resource theories

Proves theorems on resource convertibility and interconversion

Provides a basis for identifying universal features of resource theories

Abstract

In many different fields of science, it is useful to characterize physical states and processes as resources. Chemistry, thermodynamics, Shannon's theory of communication channels, and the theory of quantum entanglement are prominent examples. Questions addressed by a theory of resources include: Which resources can be converted into which other ones? What is the rate at which arbitrarily many copies of one resource can be converted into arbitrarily many copies of another? Can a catalyst help in making an impossible transformation possible? How does one quantify the resource? Here, we propose a general mathematical definition of what constitutes a resource theory. We prove some general theorems about how resource theories can be constructed from theories of processes wherein there is a special class of processes that are implementable at no cost and which define the means by which the costly states and processes can be interconverted one to another. We outline how various existing resource theories fit into our framework. Our abstract characterization of resource theories is a first step in a larger project of identifying universal features and principles of resource theories. In this vein, we identify a few general results concerning resource convertibility.