Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 2 / e

The original Von Neumann entropy depends on its eigenvalues, with $0\log 0$ defined by continuity:After rank-one dephasing, the eigenvalues are the retained diagonal populations $1/2,1/4,1/4$, givingdephasing removes coherences in the specified basis, increasing the system's mixedness while preserving those measurement probabilities. It cannot be a closed-system unitary transformation, since the eigenvalues change. The subsystem entropy increase does not by itself assert an increase in the entropy of a closed system plus its environment.