Updated Kretschmann–Schlingemann–Werner (KSW) Conjecture

Establish that for all natural numbers m and for any pair of completely positive, trace-preserving linear maps Φ1, Φ2: C^{n×n} → C^{k×k} (with n, k ∈ N) and any Stinespring isometries V1, V2: C^n → C^k ⊗ C^m of Φ1 and Φ2, respectively, the inequality min_{U ∈ U(m)} ||V1 − (I ⊗ U)V2||_∞ ≤ √(2 ||Φ1 − Φ2||_⋄) holds.

Background

Stinespring’s dilation theorem ensures that every quantum channel admits many Stinespring isometries, making it natural to ask whether channel closeness (in diamond norm) implies closeness of chosen Stinespring isometries. Kretschmann–Schlingemann–Werner proved a continuity theorem guaranteeing such a relation when the environment dimension is sufficiently large, giving ||V1 − V2||∞ ≤ √(||Φ1 − Φ2||⋄) under a dimensional assumption.

This paper refines the dimensional requirement and presents a counterexample showing that dimension cannot be dropped in the original form, motivating an updated conjecture with a universal √2 factor. The author proves the conjectured bound when at least one channel has Kraus rank one (including unitary channels), but the general case remains open.

References

First — based on said example — we formulate the (updated) Kretschmann-Schlingemann-Werner conjecture:

Conjecture 1 Let completely positive, trace-preserving linear maps \Phi_1,\Phi_2:\mathbb C{n\times n}\to\mathbb C{k\times k}, n,k\in\mathbb N be given. Then for all m\in\mathbb N and for all Stinespring isometries V_1,V_2:\mathbb Cn\to\mathbb Ck\otimes\mathbb Cm of \Phi_1,\Phi_2, respectively, it holds that \min_{U\in\mathsf U(m)}|V_1-(\mathbbm1\otimes U)V_2|\infty\leq \sqrt{2|\Phi_1-\Phi_2|\diamond}.

— Progress on the Kretschmann-Schlingemann-Werner Conjecture  (2308.15389 - Ende, 2023) in Conjecture 1, Introduction (Eq. (conj_KSW08_update))