bibkey: vomende2024diagonalizable
authors: Frederik vom Ende
year: 2024
title: "Almost All Quantum Channels Are Diagonalizable"
doi: 10.1142/S1230161224500124
url: https://arxiv.org/abs/2403.19643v3
claim: "The paper shows that diagonalizable maps are dense in several analytically path-connected classes of channels once one element with only simple eigenvalues is found (Lemma lemma_hartfiel), and leaves as an open problem a construction, valid for all Gibbs states, of Gibbs-stochastic matrices with distinct eigenvalues."
strata_touched:
  - D5/S3/Quantum/Matrix/GibbsStochasticSimpleSpectrum
license: citation-only
triage: anchor

Almost all quantum channels are diagonalizable

F. vom Ende, arXiv:2403.19643v3 [quant-ph]; Open Systems & Information Dynamics 31 (2024). Quotations are from the v3 TeX source.

The density lemma (Lemma lemma_hartfiel):

Given any finite-dimensional vector space $\mathcal{V}$ and any subset $\mathcal{C}\subseteq \mathcal{L}(\mathcal{V})$ which is analytically path connected (i.e.~for all $X_1,X_2\in \mathcal{C}$ there exists a real-analytic function $\gamma :[0,1]\to \mathcal{C}$ such that $\gamma (0)=X_1$, $\gamma (1)=X_2$) the following statements are equivalent. (i) There exists $Z\in \mathcal{C}$ such that $Z$ has only simple eigenvalues. (ii) $\overline{\{Y\in \mathcal{C}:Y\text{ has only simple eigenvalues}\}}=\mathcal{C}$.

(the two items are written on one line here).

The open problem (Outlook, with its footnote):

for the case of thermal operations, for example, one would have to first show that almost all Gibbs-stochastic matrices have only simple eigenvalues—regardless of the exact Gibbs state

For those interested: The last point is what makes this non-trivial. While the $\beta$-permutations \cite{Alhambra19,Mazurek19,PolytopeDegen22} are candidates for Gibbs-stochastic matrices with distinct eigenvalues many of them are only valid in certain parameter regimes, i.e.~only if the eigenvalues of the Gibbs state are interrelated in a specific way. We leave a general construction that is valid for all Gibbs states as an open problem.

Verified locator

Source: https://arxiv.org/abs/2403.19643v3

• URL: https://arxiv.org/abs/2403.19643v3 (source channel_diag_approx.tex, md5 18bc93c300903edbf34bf142ad00519d):
  • Lemma lemma_hartfiel, l. 170–181;
  • the open problem and its footnote, l. 509–513.
• DOI: 10.1142/S1230161224500124 (https://doi.org/10.1142/S1230161224500124; Crossref lists Open Systems & Information Dynamics, volume 31, issue 3, published 26 August 2024).