Progress on the Kretschmann-Schlingemann-Werner Conjecture

• URL: http://arxiv.org/abs/2308.15389v3
• Date: Thu, 21 Dec 2023 17:18:35 GMT
• Title: Progress on the Kretschmann-Schlingemann-Werner Conjecture
• Authors: Frederik vom Ende
• Abstract summary: Given any pair of quantum channels $Phi_1,Phi$ such that at least one of them has Kraus rank one, we prove that there exists a unitary $U$ on the environment such that $V_(bf1otimes U)V|_inftyleqsqrt2|Phi|_diamond$. 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 quantum channels.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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