Progress on the Kretschmann-Schlingemann-Werner Conjecture

TL;DR

This paper proves a bound relating the difference of quantum channels with their Stinespring isometries when one channel has Kraus rank one, confirming a conjecture and establishing optimal constants.

Contribution

It establishes a new inequality connecting channel differences and isometry differences, confirming the Kretschmann-Schlingemann-Werner conjecture in a specific case.

Findings

Proved the inequality with optimal constant for channels with Kraus rank one.

Provided an example demonstrating the optimality of the constant.

Conjecture that the inequality holds universally for all channel pairs.

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.