Progress on the Kretschmann-Schlingemann-Werner Conjecture

Given any pair of quantum channels ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2$ 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 $\mathrm{\parallel }{V}_1-(\mathbf{1}\otimes U)V_2{\mathrm{\parallel }}_{\mathrm{\infty }}\le \sqrt{2\mathrm{\parallel }{\mathrm{\Phi }}_1-{\mathrm{\Phi }}_2{\mathrm{\parallel }}_{\diamond }}$. Moreover, we provide a simple example which shows that the factor $\sqrt{2}$ on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.