Spectral Analysis for Gaussian Quantum Markov Semigroups
Franco Fagnola, Zheng Li
TL;DR
The paper analyzes the spectrum of the induced generator $L^{(s)}$ of a Gaussian quantum Markov semigroup on the Hilbert–Schmidt space, under a faithful invariant state and without assuming detailed balance. It shows that all eigenvalues are governed by the drift matrix $\boldsymbol{Z}$, with base eigenvalues from $\boldsymbol{Z}$ and all eigenvalues expressible as $n\lambda+m\mu$, where $\lambda,\mu$ are drift-eigenvalues; the adjoint $L^{(s)*}$ shares the same spectral pattern. A quasi-derivation property is leveraged to generate higher-order eigenvalues from the base ones, and two natural embeddings (KMS with $s=1/2$ and GNS with $s=0$) are treated to relate the spectra of $L^{(s)}$ and $L^{(s)*}$. The authors prove that if $L^{(s)}$ has a spectral gap, then both $L^{(s)}$ and $L^{(s)*}$ have compact resolvents and their spectra consist entirely of the identified eigenvalues, with the KMS gap dominating the GNS gap in the diagonal invariant-state case. The work extends classical Ornstein–Uhlenbeck spectral insights to the quantum, unbounded-setting and clarifies the role of the invariant state in spectral properties of Gaussian QMSs.
Abstract
We investigate the spectrum of the generator induced on the space of Hilbert-Schmidt operators by a Gaussian quantum Markov semigroup with a faithful normal invariant state in the general case, without any symmetry or quantum detailed balance assumptions. We prove that the eigenvalues are entirely determined by those of the drift matrix, similarly to classical Ornstein-Uhlenbeck semigroups. This result is established using a quasi-derivation property of the generator. Moreover, the same spectral property holds for the adjoint of the induced generator. Finally, we show that these eigenvalues constitute the entire spectrum when the induced generator has a spectral gap.