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A microscopic definition of the thermodynamic entropy in an isolated quantumsystem must satisfy (i) additivity, (ii) extensivity and (iii) the second lawof thermodynamics. We show that the diagonal entropy, which is the Shannonentropy in the energy eigenbasis at each instant of time, meets the first tworequirements and that the third requirement is satisfied if an arbitraryexternal operation is performed at typical times. In terms of the diagonalentropy, thermodynamic irreversibility follows from the facts that theHamiltonian dynamics restricts quantum trajectories under unitary evolution andthat the external operation is performed without referring to any particularinformation about the microscopic state of the system.

The Second Law of Thermodynamics under Unitary Evolution and External Operations