Can quantum theory be characterized in information-theoretic terms?

Does information play a significant role in the foundations of physics? We investigate whether information-theoretic constraints characterize quantum theory. In a C*-algebraic framework, this is known to hold via three equivalences: no broadcasting and noncommutativity; no bit commitment and nonlocality; no signalling and kinematic independence. But this complex linear framework could be said to build in quantum theory from the start. We show that the first two equivalences break, and the third holds, in a framework of generalized, possibly nonlinear, C*-algebras. This uncovers a hierarchy of notions of when (quantum) information is classical.