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Entropy, majorization, and thermodynamics in general probabilistic theories.

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Much progress has recently been made on the fine-grained thermodynamics and statistical mechanics of microscopic physical systems, by conceiving of thermodynamics as a resource theory: one which governs which transitions between states are possible using specified "thermodynamic" (e.g. adiabatic or isothermal) means. In this talk we lay some groundwork for investigating thermodynamics in generalized probabilistic theories. We describe simple, but fairly strong, postulates: unique spectrality, projectivity, and symmetry of transition probabilities, that imply that a system has a well-behaved convex analogue of the spectrum, and show that the spectrum of a state majorizes the outcome probabilities of any fine-grained measurement, allowing the operationally defined measurement entropy (and Schur-concave analogues) to be calculated from the spectrum. These are implied by, but probably weaker than, Axioms 1 (weak spectrality) and 2 (strong symmetry) of a recent characterization of Jordan-algebraic and quantum systems by Barnum, Mueller, and Ududec. It is an open question whether theories beyond the Jordan-algebraic ones satisfy them. We describe how part of von Neumann's argument that spectral entropy is a good candidate for thermodynamic entropy generalizes to systems satisfying our postulates, and discuss whether its assumptions are reasonable there, suggesting that the extendibility of certain processes to reversible ones is crucial. We will discuss further postulates and results that might suffice to obtain, in this more general setting, a thermodynamical resource theory similar to the one that is emerging for quantum theory.

(Joint work with Jon Barrett, Marius Krumm, Matt Leifer, Markus Mueller.)