Second quantization and compactness of induced Gaussian quantum Markov semigroups
Abstract
We characterize compactness of the Hilbert-Schmidt semigroups induced by finite-mode Gaussian quantum Markov semigroups with a faithful normal invariant state. A unitary second-quantization representation for every $ s $-embedding shows that compactness at one positive time, compact resolvent, drift stability, and uniqueness of the normal invariant state are equivalent. These conditions imply immediate trace-class membership for all embeddings. We derive exact trace-norm and Hilbert-Schmidt norm formulas, an embedding-independent trace identity, and the generator spectrum with algebraic multiplicities. Examples show that compactness does not require the existence of a spectral gap and that a spectral gap need not imply compactness when the invariant state is not unique.
Related
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BibTeX
@misc{li2026second,
title = {{Second quantization and compactness of induced Gaussian quantum Markov semigroups}},
author = {Zheng Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13358},
eprint = {2610.13358},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13358},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Second quantization and compactness of induced Gaussian quantum Markov semigroups
AU - Zheng Li
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.13358
UR - https://arxiv.org/abs/2610.13358
AB - We characterize compactness of the Hilbert-Schmidt semigroups induced by finite-mode Gaussian quantum Markov semigroups with a faithful normal invariant state. A unitary second-quantization representation for every $ s $-embedding shows that compactness at one positive time, compact resolvent, drift stability, and uniqueness of the normal invariant state are equivalent. These conditions imply immediate trace-class membership for all embeddings. We derive exact trace-norm and Hilbert-Schmidt norm formulas, an embedding-independent trace identity, and the generator spectrum with algebraic multiplicities. Examples show that compactness does not require the existence of a spectral gap and that a spectral gap need not imply compactness when the invariant state is not unique.
ER -