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.
Let
(Tt)t≥0 be a Gaussian quantum Markov semigroup with a faithful normal invariant state
ρ. For every
s∈[0,1], the
s-embedding associated with
ρ induces a contraction semigroup
(Tt(s))t≥0 on the Hilbert--Schmidt space
B2(h); let
L(s) denote its generator. Without assuming symmetry or quantum detailed balance, we determine the full spectra of
L(s) and
L(s)∗: they consist, respectively, of the non-negative integer combinations of the eigenvalues of the phase-space drift matrix and of their complex conjugates. We also diagonalize the self-adjoint closure of
L(s)∗+L(s) and obtain an explicit formula for the spectral gap. Using quantum characteristic functions, we represent
Tt(s), up to unitary equivalence, as a complex Gaussian integral operator and prove that its kernel is square-integrable for every
t>0. Hence
Tt(s) is a Hilbert--Schmidt operator on
B2(h) for every
t>0, and every induced semigroup is immediately compact. Consequently,
L(s) and
L(s)∗ have compact resolvent for all
s∈[0,1], and their point spectra exhaust their full spectra. These results provide a full quantum counterpart of classical Ornstein--Uhlenbeck spectral theory.
We study regularisation and irreducibility properties of Gaussian quantum Markov semigroups (GQMSs) acting on continuous-variable quantum systems. We first identify a natural notion of regularity for operators in this setting, which allows us to formulate and characterise the smoothing effects of GQMSs in terms of algebraic conditions involving the drift and quantum diffusion matrices. These conditions establish a connection with the controllability theory of quantum linear systems and with the structure of decoherence-free subsystems.
We then characterise irreducibility through several equivalent algebraic criteria: one formulated in terms of the drift and quantum diffusion matrices, one in terms of the operators appearing in the generalised GKLS representation of the generator, and a third given by a quantum analogue of Hörmander's condition. A central and somewhat surprising consequence is that, in contrast with the classical case, irreducibility is strictly stronger than conditions ensuring regularisation. Our results provide an algebraic framework for analysing these properties and lay the groundwork for a broader study of reducible Gaussian quantum Markov semigroups and, more generally, more general relevant quantum Markov semigroups on continuous-variable systems.
Gaussian quantum Markov semigroups are the natural non-commutative extension of classical Ornstein-Uhlenbeck semigroups. They arise in open quantum systems of bosons where canonical non-commuting random variables of positions and momenta come into play. If there exists a faithful invariant density we explicitly compute the optimal exponential convergence rate, namely the spectral gap of the generator, in non-commutative L 2 spaces determined by the invariant density showing that the exact value is the lowest eigenvalue of a certain matrix determined by the diffusion and drift matrices. The spectral gap turns out to depend on the non-commutative L 2 space considered, whether the one determined by the so-called GNS or KMS multiplication by the square root of the invariant density. In the first case, it is strictly positive if and only if there is the maximum number of linearly independent noises. While, we exhibit explicit examples in which it is strictly positive only with KMS multiplication. We do not assume any symmetry or quantum detailed balance condition with respect to the invariant density.
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.
The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is represented in a generalized GKLS form with an operator [Formula: see text] quadratic in creation and annihilation operators and Kraus operators [Formula: see text] linear in creation and annihilation operators. Kraus operators, commutators [Formula: see text] and iterated commutators [Formula: see text] up to the order [Formula: see text], as linear combinations of creation and annihilation operators determine a vector in [Formula: see text]. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate [Formula: see text], under the technical condition that the domains of [Formula: see text] and the number operator coincide. Conversely, we show that this condition is also necessary if the linear space generated by Kraus operators and their iterated commutator with [Formula: see text] is fully non-commutative.