Trace-class spectra of irreducible Gaussian quantum Markov semigroups
Abstract
We determine the spectrum of the predual generator of an irreducible Gaussian quantum Markov semigroup on trace-class operators, the natural space for density operators. The drift matrix governs the linear part of the Gaussian evolution and organizes the spectral classification. When the linear flow generated by this matrix is not periodic, the spectrum is the closed left half-plane. If all drift eigenvalues have negative real part, every interior point is an eigenvalue. There are no trace-class eigenvectors if some drift eigenvalue has positive real part or all have zero real part. For periodic flow, the spectrum consists of unions of horizontal half-lines or filled parabolic regions. Examples show that irreducibility cannot in general be omitted. We apply the results to oscillators, free particles, and boson chains.
Related
- Second quantization and compactness of induced Gaussian quantum Markov semigroups
- Entropy Contraction and Hypercontractivity for Gaussian Quantum Markov Semigroups
- Irreducibility and regularisation properties of Gaussian quantum Markov semigroups
- Quantum Cheeger Inequalities for KMS-Symmetric Quantum Markov Semigroups
- Some entropic inequalities for primitive KMS symmetric quantum Markov semigroups
BibTeX
@misc{fagnola2026traceclass,
title = {{Trace-class spectra of irreducible Gaussian quantum Markov semigroups}},
author = {Franco Fagnola and Zheng Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06957},
eprint = {2609.06957},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06957},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Trace-class spectra of irreducible Gaussian quantum Markov semigroups
AU - Franco Fagnola
AU - Zheng Li
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.06957
UR - https://arxiv.org/abs/2609.06957
AB - We determine the spectrum of the predual generator of an irreducible Gaussian quantum Markov semigroup on trace-class operators, the natural space for density operators. The drift matrix governs the linear part of the Gaussian evolution and organizes the spectral classification. When the linear flow generated by this matrix is not periodic, the spectrum is the closed left half-plane. If all drift eigenvalues have negative real part, every interior point is an eigenvalue. There are no trace-class eigenvectors if some drift eigenvalue has positive real part or all have zero real part. For periodic flow, the spectrum consists of unions of horizontal half-lines or filled parabolic regions. Examples show that irreducibility cannot in general be omitted. We apply the results to oscillators, free particles, and boson chains.
ER -