  In quantum theory every state can be diagonalized, i.e. decomposed as a
convex combination of perfectly distinguishable pure states. This elementary
structure plays an ubiquitous role in quantum mechanics, quantum information
theory, and quantum statistical mechanics, where 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 occurring with unit probability on
some state. For theories satisfying our four axioms, we show a constructive
algorithm for diagonalizing every given state. The diagonalization result
allows us to formulate a majorization criterion that captures the
convertibility of states in the operational resource theory of purity, where
random reversible transformations are regarded as free operations.
