Quantum microcanonical entropy of a pair of observables

The quantum analogue of the classical theory of the joint microcanonical entropy of a pair of observables is investigated for a system of a large number of identical non-interacting subsystems. It is shown that the quantum joint entropy coincides with the classical joint entropy of an appropriately chosen auxiliary classical system, and known results for classical systems are applied to prove the equivalence of the quantum microcanonical and quantum canonical ensembles.