A dimension-independent strict submultiplicativity for the transposition map in diamond norm

Abstract

We prove that there exists an absolute constant $\alpha <1$ such that for every finite dimension $d$ and every quantum channel $T$ on $\mathsf{L}({\mathbb{C}}^d)$, ${\Vert \Theta \circ (\mathrm{id}-T)\Vert }_{\diamond }\le \alpha {\Vert \Theta \Vert }_{\diamond }{\Vert \mathrm{id}-T\Vert }_{\diamond }$, where $\Theta$ is the transposition map. In fact we show the explicit choice $\alpha =1/\sqrt{2}$ works.

6 pages