The CBH characterisation theorem beyond algebraic quantum theory

TL;DR

This paper generalizes the CBH theorem beyond C*-algebras, revealing a more nuanced hierarchy of classical and quantum notions by categorically extending the framework and analyzing the mathematical structures underlying quantum theory.

Contribution

It extends the CBH characterization theorem to a categorical setting, clarifying the role of complex linearity and uncovering a hierarchy of classicality and quantumness.

Findings

Equivalence (1) remains valid in the generalized framework

Implication (2) becomes a strict implication

Implication (3) generally fails in the categorical setting

Abstract

The CBH theorem characterises quantum theory within a C*-algebraic framework. Namely, mathematical properties of C*-algebras modelling quantum systems are equivalent to constraints that are information-theoretic in nature: (1) noncommutativity of subalgebras is equivalent to impossibility of signalling; (2) noncommutativity of the whole algebra is equivalent to impossibility of broadcasting; (3) the existence of entangled states is implied by the impossibility of secure bit commitment (with the converse conjectured). However, the C*-algebraic framework has drawn criticism as it already contains much of themathematical structure of quantum theory such as complex linearity. We address this issue by a generalising C*-algebras categorically. In this framework, equivalence (1) holds, equivalence (2) becomes a strict implication, and implication (3) fails in general. Thus we identify exactly what work is being done by the complex-linear structure of C*-algebras. In doing so, we uncover a richer hierarchy of notions of 'classicality' and 'quantumness' of information than visible in the concrete case.