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Progress on the Kretschmann-Schlingemann-Werner Conjecture
[Submitted on 29 Aug 2023 (v1), last revised 18 Jun 2026 (this v · 2026-06-19 · via math updates on arXiv.org

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Abstract: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 $\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|\Phi_1-\Phi_2\|_\diamond}$. 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.

Submission history

From: Frederik Vom Ende [view email]
[v1] Tue, 29 Aug 2023 15:32:58 UTC (94 KB)
[v2] Tue, 5 Sep 2023 14:47:53 UTC (18 KB)
[v3] Thu, 21 Dec 2023 17:18:35 UTC (18 KB)
[v4] Thu, 18 Jun 2026 08:29:40 UTC (18 KB)