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The second law of thermodynamics for pure quantum states

A version of the second law of thermodynamics states that one cannot lower the energy of an isolated system by a cyclic operation. We prove this law without introducing statistical ensembles and by resorting only to quantum mechanics. We choose the initial state as a pure quantum state whose energy is almost E_0 but not too sharply concentrated at energy eigenvalues. Then after an arbitrary unitary time evolution which follows a typical "waiting time", the probability of observing the energy lower than E_0 is proved to be negligibly small.