Progress on the Kretschmann-Schlingemann-Werner Conjecture (pp1319-1330) 
          Frederik vom Ende 
          doi: https://doi.org/10.26421/QIC23.15-16-5
Abstracts: 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 $\|V_1-(\mathbbm1\otimes U)V_2\|_\infty\leq\sqrt{2\|\Phi_1-\Phi_2\|_\diamond}$. 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.
Key Words: Stinespring dilation, Stinespring isometries, quantum channels, diamond norm