Diagonalising states: an operational route towards a resource theory of purity

In quantum theory every state can be diagonalised, i.e. decomposed as a convex combination of perfectly distinguishable pure states. This fact is crucial in quantum statistical mechanics, as it provides the foundation for the notions of majorisation and entropy. A natural question then arises: can we give an operational characterisation of them? We address this question in the framework of general probabilistic theories, presenting a set of axioms that guarantee that every state can be diagonalised: Causality, Purity Preservation, Purification, and Pure Sharpness. If we add the Permutability and Strong Symmetry axioms, which are in fact completely equivalent in theories satisfying the other axioms, the diagonalisation result allows us to define a well-behaved majorisation preorder on states. Indeed this majorisation criterion fully captures the convertibility of states in the operational resource theory of purity where random reversible transformations are regarded as free operations. One can also put forward two alternative notions of purity as a resource: one where free operations are unital channels, and another where free operations are generated by reversible interactions with an environment in the invariant state. Under the validity of the above axioms, all these definitions are in fact equivalent, i.e. they all lead to the same preorder on states, which is given by majorisation, in the very same way as in quantum theory.