  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.
