 Solution

Let $\Delta =\rho -\sigma ={\Delta }_{+}-{\Delta }_{-}$ be its positive-negative decomposition. Since $\mathrm{Tr}\Delta =0$, the positive and negative parts have equal trace, namely $D(\rho ,\sigma )$. For every positive contraction $M$,Here both products have nonnegative trace, and $M\le I$ bounds the first term. Equality is attained by the orthogonal projection $P_{+}$ onto the positive spectral subspace of $\Delta$, since $P_{+}{\Delta }_{+}={\Delta }_{+}$ and $P_{+}{\Delta }_{-}=0$. Thus the variational characterization of trace distance is