Operational axioms for diagonalizing states

Understand

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.