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Progress on the Kretschmann-Schlingemann-Werner Conjecture

Frederik vom Ende

TL;DR

The paper addresses the Kretschmann-Schlingemann-Werner conjecture, investigating how close two quantum channels are in the diamond norm relative to how close their Stinespring dilations can be made via a shared environment. Using the operational fidelity (Bures distance) and a two-case analysis, it proves the conjecture in the Kraus rank one case, showing $\min_{W} \|V_1-(\mathbb{1}\otimes W)V_2\|_\infty \le \sqrt{2\|\Phi_1-\Phi_2\|_\diamond}$, and demonstrates the necessity of the $\sqrt{2}$ factor through explicit examples. The work also clarifies the role of the environment dimension, proving a precise relation when the Kraus ranks are favorable and illustrating that a universal finite-dimensional bound cannot beat the $\sqrt{2}$ constant. These results provide a partial resolution of the conjecture and motivate future work to extend to general channels and to infinite-dimensional or dynamic settings.

Abstract

Given any pair of quantum channels $Φ_1,Φ_2$ 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 $\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond}$. 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.

Progress on the Kretschmann-Schlingemann-Werner Conjecture

TL;DR

The paper addresses the Kretschmann-Schlingemann-Werner conjecture, investigating how close two quantum channels are in the diamond norm relative to how close their Stinespring dilations can be made via a shared environment. Using the operational fidelity (Bures distance) and a two-case analysis, it proves the conjecture in the Kraus rank one case, showing , and demonstrates the necessity of the factor through explicit examples. The work also clarifies the role of the environment dimension, proving a precise relation when the Kraus ranks are favorable and illustrating that a universal finite-dimensional bound cannot beat the constant. These results provide a partial resolution of the conjecture and motivate future work to extend to general channels and to infinite-dimensional or dynamic settings.

Abstract

Given any pair of quantum channels such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries , we prove that there exists a unitary on the environment such that . Moreover, we provide a simple example which shows that the factor on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.
Paper Structure (3 sections, 1 theorem, 38 equations, 1 table)

This paper contains 3 sections, 1 theorem, 38 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Conclusions and Outlook

Key Result

Lemma 1

Let $m,n,k\in\mathbb N$, $U\in\mathbb C^{k\times n}$, a vector $\phi\in\mathbb C^m$ with $\|\phi\|=1$, and a linear map $V:\mathbb C^n\to\mathbb C^k\otimes\mathbb C^m$ be given such that $U$ and $V$ are isometries. If there exists $W\in\mathsf U(m)$ such that $\|U\otimes|\phi \rangle-(\mathbbm1\otim

Theorems & Definitions (1)

  • Lemma 1