Progress on the Kretschmann-Schlingemann-Werner Conjecture

Abstract: Given any pair of quantum channels $\Phi_1,\Phi_2$ such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries $V_1,V_2$, we prove that there exists a unitary $U$ on the environment such that $|V_1-({\bf1}\otimes U)V_2|\infty\leq\sqrt{2|\Phi_1-\Phi_2|\diamond}$. Moreover, we provide a simple example which shows that the factor $\sqrt2$ on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.