  Von Neumann obtained the formula for the entropy of a quantum state by
assuming the validity of the second law of thermodynamics in a thought
experiment involving semipermeable membranes and an ideal gas of
quantum-labeled particles. Despite being operational in the most part, von
Neumann's argument crucially departs from an operational narrative in its use
of the spectral theorem. In this work we show that the role of the spectral
theorem in von Neumann's argument can be taken over by the operational
assumptions of repeatability and reversibility, and using these we are able to
explore the consequences of the second law also in theories that do not possess
a unique spectral decomposition. As a byproduct, we obtain the
Groenewold--Ozawa information gain as a natural monotone for a suitable
ordering of instruments, providing it with an operational interpretation valid
in quantum theory and beyond.
