bibkey: chruscinskidenisovtarnowskizyczkowski2025spectral
authors: Dariusz Chruściński; Sergey Denisov; Wojciech Tarnowski; Karol Życzkowski
year: 2025
title: "Spectral delineation of Markov generators: classical vs quantum"
doi: 10.1088/1751-8121/ae58d2
url: https://arxiv.org/abs/2504.07903v1
claim: "The paper rescales generators of classical and quantum Markov semigroups so that their spectra lie in the disk of radius one centred at -1, proves such confinement for purely dissipative GKLS generators under a quantum rescaling parameter, and conjectures (Conjecture 3) that for generators of positive trace-preserving dynamics the scale given by the maximal absolute diagonal value on pure states already confines the spectrum."
strata_touched:
  - D5/S3/Quantum/Dynamics/DissipatorSpectralDiskRefutation
license: citation-only
triage: anchor

Spectral delineation of Markov generators: classical vs quantum

D. Chruściński, S. Denisov, W. Tarnowski and K. Życzkowski, arXiv:2504.07903v1 (2025); published in J. Phys. A 59 (2026) as “Corralling spectra of generators of time-continuous Markov evolution (both classical and quantum)”. Quotations are from the v1 TeX source.

The GKLS form:

$\mathcal{L}(\rho )=-i[H,\rho ]+{\sum }_k{\gamma }_k(L_k\rho L_k^{†}-\frac{1}{2}\{L_k^{†}{L}_k,\rho \}),$

where $H$ is an Hermitian operator (effective system’s Hamiltonian), $L_k$ are the so-called jump operators, and ${\gamma }_k>0$ are the damping rates.

The canonical form of a purely dissipative generator:

$\mathcal{L}(\rho )={\sum }_{\alpha }{\gamma }_{\alpha }\left(L_{\alpha }\rho L_{\alpha }^{†}-\frac{1}{2}\{L_{\alpha }^{†}{L}_{\alpha },\rho \}\right)$

with $\mathrm{Tr}{L}_{\alpha }=0$, and $\mathrm{Tr}(L_{\alpha }^{†}{L}_{\beta })={\delta }_{\alpha \beta }$.

Conjecture 3:

If $\mathcal{L}$ is a generator of positive trace-preserving dynamics and ${\nu }_{\text{q},3}:={\max }_{\Vert \psi \Vert =1}|⟨\psi |\mathcal{L}(|\psi ⟩⟨\psi |)|\psi ⟩|$, then the spectrum of $\frac{1}{{\nu }_{\text{q},3}}\mathcal{L}$ is confined in the unit disk $\mathbb{D}(-1,1)$.

Verified locator

Source: https://arxiv.org/abs/2504.07903v1

• URL: https://arxiv.org/abs/2504.07903v1 (source bounding_6.tex, md5 31cb6e7a93cfb4dac99f83105a784d57):
  • the GKLS form, l. 205–209;
  • purely dissipative generators and the disk $\mathbb{D}(-1,1)$, l. 224–227;
  • the canonical traceless form, l. 794–799;
  • Conjecture 3, l. 1044–1050.
• DOI: 10.1088/1751-8121/ae58d2 (https://doi.org/10.1088/1751-8121/ae58d2).