= Trace-distance contraction under quantum channels
{title2=$D(\mathcal E(\rho),\mathcal E(\sigma))\leq D(\rho,\sigma)$}
By the <variational characterization of trace distance>, maximize $\operatorname{Tr}[P\mathcal E(\rho-\sigma)]$ over $0\leq P\leq I$. The adjoint of a <quantum channel> is positive and unital, so $0\leq\mathcal E^*(P)\leq I$. 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>.
Back to article page
