Operational axioms for diagonalizing states

Abstract

In quantum theory every state can be diagonalized, i.e. decomposed as a convex combination of perfectly distinguishable pure states. This feature is crucial for quantum statistical mechanics, as it provides the foundation for the notions of majorization and entropy. A natural question then arises: can we reconstruct these notions from purely operational axioms? We address this question in the framework of general probabilistic theories, presenting a set of axioms that guarantee that every state can be diagonalized. The first axiom is Causality, which ensures that the marginal of a bipartite state is well defined. Then, Purity Preservation states that the set of pure transformations is closed under composition. The third axiom is Purification, which allows to assign a pure state to the composition of a system with its environment. Finally, we introduce the axiom of Pure Sharpness, stating that for every system there exists at least one pure effect that occurs with unit probability on some state. Using these axioms, we present a constructive method to decompose every given state into perfectly distinguishable pure states.