Wishart cones and quantum geometry

TL;DR

This paper explores the connection between invariant cones in von Neumann algebras, quantum information geometry, and quantum field theory, revealing new links between these mathematical and physical frameworks.

Contribution

It explicitly relates Connes--Araki--Haagerup cones to Wishart laws and information geometry, bridging quantum algebra and quantum geometry.

Findings

Connes--Araki--Haagerup cones are related to Wishart laws.

New connections between quantum information geometry and quantum field theory.

Highlights the role of invariant cones in quantum geometry.

Abstract

An important object appearing in the framework of the Tomita--Takesaki theory is an invariant cone under the modular automorphism group of von Neumann algebras. As a result of the connection between von Neumann algebras and quantum field theory, von Neumann algebras have become increasingly important for (higher) category theory and topology. We show explicitly how an example of a class of cones discovered by Connes--Araki--Haagerup (CAH), invariant under the modular automorphism group, are related to Wishart laws and information geometry. Given its relation to 2D quantum field theory this highlights new relations between (quantum) information geometry and quantum geometry.