Extending Resource Monotones using Kan Extensions

Abstract

In this paper we generalize the framework proposed by Gour and Tomamichel regarding extensions of monotones for resource theories. A monotone for a resource theory assigns a real number to each resource in the theory signifying the utility or the value of the resource. Gour and Tomamichel studied the problem of extending monotones using set-theoretical framework when a resource theory embeds fully and faithfully into the larger theory. One can generalize the problem of computing monotone extensions to scenarios when there exists a functorial transformation of one resource theory to another instead of just a full and faithful inclusion. In this article, we show that (point-wise) Kan extensions provide a precise categorical framework to describe and compute such extensions of monotones. To set up monotone extensions using Kan extensions, we introduce partitioned categories (pCat)as a framework for resource theories and pCat functors to formalize relationship between resource theories. We describe monotones as pCat functors into the preorder of non-negative real numbers, and describe extending monotones along any pCat functor using Kan extensions. We show how our framework works by applying it to extend entanglement monotones for bipartite pure states to bipartite mixed states, to extend classical divergences to the quantum setting, and to extend a non-uniformity monotone from classical probabilistic theory to quantum theory.

arXiv commentIn Proceedings ACT 2022, arXiv:2307.15519

References in corpus (22)

• Quantum entanglement
• Quantum Coherence as a Resource
• Quantum Resource Theories
• The role of quantum information in thermodynamics --- a topical review
• From Classical to Quantum Shannon Theory
• A Schmidt number for density matrices
• The resource theory of informational nonequilibrium in thermodynamics
• The general structure of quantum resource theories
• A mathematical theory of resources
• (Quantumness in the context of) Resource Theories
• An introductory review of the resource theory approach to thermodynamics
• A Characterization of Entropy in Terms of Information Loss
• Entropy and Information Causality in General Probabilistic Theories
• Entropy in general physical theories
• Distinguishability Measures and Entropies for General Probabilistic Theories
• Quantum Relative Lorenz Curves
• Dynamical Entanglement
• Relative submajorization and its use in quantum resource theories
• Thermodynamics and the structure of quantum theory
• Optimal Extensions of Resource Measures and their Applications
• Uniqueness and Optimality of Dynamical Extensions of Divergences
• Information-theoretic foundations of thermodynamics in general probabilistic theories

Cited by in corpus (1)

• Thermodynamic state convertibility is determined by qubit cooling and heating