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 $\mathsf{L}(\mathbb{C}^d)$, $\left\|\Theta\circ(\mathrm{id}-T)\right\|_\diamond \le \alpha\,\left\|\Theta\right\|_\diamond\,\left\|\mathrm{id}-T\right\|_\diamond$, where $\Theta$ is the transposition map. In fact we show the explicit choice $\alpha=1/\sqrt{2}$ works.