The CBH characterisation theorem beyond algebraic quantum theory

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