Trace-distance contraction under quantum channels ($D(\mathcal{E}(\rho ),\mathcal{E}(\sigma ))\le D(\rho ,\sigma )$)

By the variational characterization of trace distance, maximize $\mathrm{Tr}[P\mathcal{E}(\rho -\sigma )]$ over $0\le P\le I$. The adjoint of a quantum channel is positive and unital, so $0\le {\mathcal{E}}^{\ast }(P)\le I$. Moving the channel onto $P$ therefore restricts, rather than enlarges, the effects available in the input variational problem. This proves the displayed bound for every pair of density operators.