Abstract
Given any pair of quantum channels Φ₁,Φ₂ such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries V₁,V₂, we prove that there exists a unitary U on the environment such that V₁-(1⊗ U)V₂_∞≤√2Φ₁-Φ₂. Moreover, we provide a simple example which shows that the factor √2 on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.