Progress on the Kretschmann-Schlingemann-Werner Conjecture

Abstract

Given any pair of quantum channels $\hat{I}{¦}_1,\hat{I}{¦}_2$ 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 $\Vert V_1-(1\otimes U)V_2{\Vert }_{\infty }\le \sqrt{2\Vert \hat{I}{¦}_1-\hat{I}{¦}_2{\Vert }_{\diamond }}$. 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.

9+3 pages. UPDATE: conjecture proven in arXiv:2606.16418