bibkey: roofehkarimipour2024wernerholevo
authors: Shayan Roofeh; Vahid Karimipour
year: 2024
title: "Noisy Werner-Holevo channel and its properties"
doi: 10.1103/PhysRevA.109.052620
url: https://arxiv.org/abs/2310.15353v7
claim: "The paper studies the qutrit channel Λ_x = (1-x) id + x Λ_WH, computes its spectrum and determinant (negative for 2/3 < x ≤ 1, so not infinitely divisible there), and leaves open whether Λ_x is the exponential of Lindbladian dynamics for 0 ≤ x ≤ 2/3."
strata_touched:
  - D5/S3/Quantum/QuantumChannels/NoisyWernerHolevoLindbladEmbedding
license: citation-only
triage: anchor

Noisy Werner–Holevo channel and its properties

S. Roofeh and V. Karimipour, arXiv:2310.15353v7 [quant-ph]; Phys. Rev. A 109, 052620 (2024). Quotations are from the v7 TeX source.

The channel:

${\Lambda }_x(\rho )=(1-x)\rho +x{\Lambda }_{{}_{WH}}(\rho )=(1-x)\rho -\frac{x}{2}{\rho }^T+\frac{x}{2}\mathrm{tr}(\rho )\mathbb{I}\text{h}0\le x\le 1,$

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The spectrum and the question:

Counting the degeneracies of the eigenvalues, this shows that the determinant of the channel is equal to $Det({\Lambda }_x)=(1-\frac{x}{2}{)}^3(1-\frac{3}{2}x{)}^5.$ This shows that the determinant of the channel is negative in a certain range, namely $Det({\Lambda }_x)<0,\text{h}\frac{2}{3}<x\le 1.$ In this range, the channel is not infinitely divisible and does not represent a Markovian evolution \cite{wolf_dividing_2008}. The question remains whether, within the range of $0\le x\le \frac{2}{3}$, the channel can be described as the exponential of Lindbladian dynamics. We will explore this question in our future work.

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Verified locator

Source: https://arxiv.org/abs/2310.15353v7

• URL: https://arxiv.org/abs/2310.15353v7 (source ModifiedLSChannel-12.tex, md5 bcff52a86196a2e1a0054a34c48a8117):
  • the channel, l. 153;
  • the spectrum, l. 236–247;
  • the question, l. 249.
• DOI: 10.1103/PhysRevA.109.052620 (https://doi.org/10.1103/PhysRevA.109.052620; Crossref lists Physical Review A, volume 109, article 052620, published 14 May 2024).