Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 65 / 3 / ii / Solution

The quantum relative entropy is $S(\rho \Vert \sigma )=\mathrm{Tr}\rho ({\log }_2\rho -{\log }_2\sigma )$ when the support of a positive operator $\rho$ is contained in that of $\sigma$, and $+\infty$ otherwise. Write spectral decompositions $\rho ={\sum }_i{r}_i|i⟩⟨i|$ and $\sigma ={\sum }_j{t}_j|u_j⟩⟨u_j|$, and set $q_i=⟨i|\sigma |i⟩$.

The weights $w_{ij}=|⟨i|u_j⟩{|}^2$ sum to one. Concavity of the scalar logarithm gives the diagonal logarithm concavity boundFor $r_i>0$, support inclusion removes overlaps with zero eigenvalues of $\sigma$; alternatively take a positive regularization and pass to the limit. The two diagonal vectors $r$ and $q$ are probability distributions. ThereforeThis proves nonnegativity of quantum relative entropy without assuming the states commute. Equality in the strict logarithmic concavity condition makes each occupied $|i⟩$ an eigenvector of $\sigma$; classical equality gives $q_i=r_i$ and no weight outside the support, hence equality occurs exactly for $\rho =\sigma$.