  We propose four information-theoretic axioms for the foundations of
statistical mechanics in general physical theories. The axioms---Causality,
Purity Preservation, Pure Sharpness, and Purification---identify a class of
theories where every mixed state can be modelled as the marginal of a pure
entangled state and where every unsharp measurement can be modelled as a sharp
measurement on a composite system. This class of theories, called sharp
theories with purification, includes quantum theory both with complex and real
amplitudes, as well as a suitable extension of classical probability theory
where classical systems can be entangled with other, non-classical systems.
Theories satisfying our axioms support well-behaved notions of majorization,
entropy, and Gibbs states, allowing for an information-theoretic derivation of
Landauer's principle. We conjecture that every theory admitting a sensible
thermodynamics must be extendable to a sharp theory with purification.
