Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 48 / 1 / a / Solution

Use logarithms to base two, so all Von Neumann entropies and Shannon entropies are measured in bits. Write the nonzero eigenvalues of ${\rho }_i$ as ${\lambda }_{ij}$. Orthogonality of the supports makes the mixture a block diagonal matrix, with eigenvalues $p_i{\lambda }_{ij}$. ThereforeEach ${\rho }_i$ has trace one. Terms with zero probability or zero eigenvalue vanish under the convention $0\log 0=0$. This proves the entropy of an orthogonal quantum mixture.