On the categorical foundations of quantum information theory: Categories and the Cramer-Rao inequality

TL;DR

This paper develops a categorical framework for quantum information theory, extending classical concepts like statistical manifolds and the Cramer-Rao inequality to quantum systems, unifying classical and quantum information.

Contribution

It introduces a novel categorical foundation for quantum information theory, integrating classical and quantum concepts and formalizing the quantum environment.

Findings

Extended statistical manifold to categories

Proposed a uniparametric Cramer-Rao inequality in the categorical setting

Unified classical and quantum information theories

Abstract

An extension of Cencov's categorical description of classical inference theory to the domain of quantum systems is presented. It provides a novel categorical foundation to the theory of quantum information that embraces both classical and quantum information theory in a natural way, while also allowing to formalise the notion of quantum environment. A first application of these ideas is provided by extending the notion of statistical manifold to incorporate categories, and investigating a possible, uniparametric Cramer-Rao inequality in this setting.