  We prove a generalization of the classic Groenewold-Lindblad entropy
inequality, combining decoherence and the quantum Bayes theorem into a simple
unified picture where decoherence increases entropy while observation decreases
it. This provides a rigorous quantum-mechanical version of the second law of
thermodynamics, governing how the entropy of a system (the entropy of its
density matrix, partial-traced over the environment and conditioned on what is
known) evolves under general decoherence and observation. The powerful tool of
spectral majorization enables both simple alternative proofs of the classic
Lindblad and Holevo inequalities without using strong subadditivity, and also
novel inequalities for decoherence and observation that hold not only for von
Neumann entropy, but also for arbitrary concave entropies.
