Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 50 / 3 / d / Solution

Take the rank-one orthogonal projections $P_i=|{\varphi }_i⟩⟨{\varphi }_i|$. Their pinching map givesThis has exactly the specified diagonal entries and no off-diagonal entries. Applying the entropy increase under nonselective projective measurement proved in part (c) therefore givesWriting $q_i=⟨{\varphi }_i|\rho |{\varphi }_i⟩$, the output Von Neumann entropy is the Shannon entropy $S({\rho }_d)=-{\sum }_i{q}_i\log q_i$. Equality holds precisely when $\rho$ was already diagonal in this basis.