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.

Functional Analysis (math.FA); Quantum Physics (quant-ph)

arXiv:2610.13358 [math.FA]

(or arXiv:2610.13358v1 [math.FA] for this version)

https://doi.org/10.48550/arXiv.2610.13358 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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