Increase of quantum volume entropy in presence of degenerate eigenenergies

The entropy of a classical thermally isolated Hamiltonian system is given by the logarithm of the measure of phase space enclosed by the constant energy hyper-surface, also known as volume entropy. It has been shown that on average the latter cannot decrease if the initial state is sampled from a classical passive distribution. Quantum extension of this result has been shown, but only for systems with a non-degenerate energy spectrum. Here we further extend to the case of possible degeneracies.