Progress on the Kretschmann-Schlingemann-Werner Conjecture

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∥∞ ≤ p2∥Φ1 − 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.