Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 4 / a / ii

Let $\Delta =\rho -\sigma ={\Delta }_{+}-{\Delta }_{-}$ be the positive-negative decomposition of a Hermitian operator. Since $\mathrm{Tr}\Delta =0$,For $0\le P\le I$,Equality is attained by the projector onto the positive eigenspace of $\Delta$. This proves the variational characterization of trace distance