A short proof of the modified Kretschmann-Schlingemann-Werner conjecture

• URL: http://arxiv.org/abs/2606.16418v1
• Date: Mon, 15 Jun 2026 08:56:32 GMT
• Title: A short proof of the modified Kretschmann-Schlingemann-Werner conjecture
• Abstract summary: We prove that there exists a unitary $U$ on $|V_1-(otimes U)V|_inftyleqsqrt2|_diamond,$ thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture.
• Score: 2.0305676256390934
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Let $Φ_1, Φ_2 : \mathbb{M}_d(\mathbb{C})\to \mathbb{M}_n(\mathbb{C})$ be two quantum channels with respective Stinespring isometries $V_1, V_2 : \mathbb{C}^{d}\to \mathbb{C}^{n} \otimes \mathbb{C}^{m}$ on any common dilation space $\mathbb{C}^{m}$. We prove that there exists a unitary $U$ on $\mathbb{C}^{m}$ such that $\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond},$ thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.

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