  Despite its enormous empirical success, the formalism of quantum theory still
raises fundamental questions: why is nature described in terms of complex
Hilbert spaces, and what modifications of it could we reasonably expect to find
in some regimes of physics? Here we address these questions by studying how
compatibility with thermodynamics constrains the structure of quantum theory.
We employ two postulates that any probabilistic theory with reasonable
thermodynamic behavior should arguably satisfy. In the framework of generalized
probabilistic theories, we show that these postulates already imply important
aspects of quantum theory, like self-duality and analogues of projective
measurements, subspaces and eigenvalues. However, they may still admit a class
of theories beyond quantum mechanics. Using a thought experiment by von
Neumann, we show that these theories admit a consistent thermodynamic notion of
entropy, and prove that the second law holds for projective measurements and
mixing procedures. Furthermore, we study additional entropy-like quantities
based on measurement probabilities and convex decomposition probabilities, and
uncover a relation between one of these quantities and Sorkin's notion of
higher-order interference.
