Generators of Gaussian QMS have a GKLS form with noise operators that are linear in creation and annihilation operators, and Hamiltonian which is quadratic. We characterize Gaussian QMS arising in the Weak Coupling Limit (WCL) of a system interacting with some reservoirs. It turns out that their generators, under some natural assumptions, have a GKLS representation with noise operators that are either annihilation or creation operators and Hamiltonian [Formula: see text] which is diagonalizable as a quadratic operator.

We develop an analytical approach to quantum Gaussian states in infinite-mode representation of the Canonical Commutation Relations (CCR's), using Yosida approximations to define integrability of possibly unbounded observables with respect to a state ρ (ρ-integrability). It turns out that all elements of the commutative *-algebra generated by a possibly unbounded ρ-integrable observable A, denoted by ⟨ A⟩, are normal and ρ-integrable. Besides, ⟨ A⟩ can be endowed with the well-defined norm ·_ρ:= tr (ρ |·| ). Our approach allows us to rigorously establish fundamental properties and derive key formulae for the mean value vector and the covariance operator. We additionally show that the covariance operator S of any Gaussian state is real, bounded, positive, and invertible, with the property that S-iJ≥ 0, being J the multiplication operator by -i on ℓ_2(ℕ).

We introduce the notion of local detailed balance for low density limit generators of quantum Markov semigroups and explore the consequences of this condition on the structure of the corresponding set of invariant states. The local detailed balance condition allows us to include the case of degenerate reference Hamiltonian and due to this degeneracy, we can compute nonequilibrium invariant states by combining the local detailed balance with generic, quasi-generic and [Formula: see text]-generic conditions, obtaining LDL Markov generators with pure invariant states.

We study decoherence-free subspaces of weak coupling limit type Markov generators and make a full description of the decoherence-free sub-algebra of the excitation energy transport AKV's model studied in Ref.(13).

We broaden the study of circulant Quantum Markov Semigroups (QMS). First, we introduce the notions of [Formula: see text]-circulant GKSL generator and [Formula: see text]-circulant QMS from the circulant case, corresponding to [Formula: see text], to an arbitrary finite group [Formula: see text]. Second, we show that each [Formula: see text]-circulant GKSL generator has a block-diagonal representation [Formula: see text], where [Formula: see text] is a [Formula: see text]-circulant matrix determined by some [Formula: see text]. Denoting by [Formula: see text] the subgroup of [Formula: see text] generated by the support of [Formula: see text], we prove that [Formula: see text] has its own block-diagonal matrix representation [Formula: see text] where [Formula: see text] is an irreducible [Formula: see text]-circulant matrix and [Formula: see text] is the index of [Formula: see text] in [Formula: see text]. Finally, we exploit such block representations to characterize the structure, steady states, and asymptotic evolution of [Formula: see text]-circulant QMSs.

We give a rigorous definition of moments of an unbounded observable with respect to a quantum state in terms of Yosida's approximations of unbounded generators of contractions semigroups. We use this notion to characterize Gaussian states in terms of the moments of the field operator, which we call Weyl moments. As a by-product, rigorous formulae for the mean value vector and the covariance matrix of a Gaussian state are obtained.

We give a rigorous definition of moments of an unbounded observable with respect to a quantum state in terms of Yosida's approximations of unbounded generators of contractions semigroups. We use this notion to characterize Gaussian states in terms of annihilation moments. As a by-product, rigorous formulae for the mean value vector and the covariance matrix of a Gaussian state are obtained.

We prove that every stationary state in the annihilator of all Kraus operators of a weak coupling limit-type Markov generator consists of two pieces, one of them supported on the interaction-free subspace and the second one on its orthogonal complement. In particular, we apply the previous result to describe in detail the structure of a slightly modified quantum transport model due to Arefeva et al. (modified AKV’s model) studied first in [J. C. García et al., Entangled and dark stationary states of excitation energy transport models in many-particles systems and photosynthesis, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21(3) (2018), Article ID: 1850018, p. 21, doi:10.1142/S0219025718500182], in terms of generalized annihilation and creation operators.

We use a natural generalization of the discrete Fourier transform to define transition maps between Hilbert subspaces and the global transport operator Z. By using these transition maps as Kraus (or noise) operators, an extension of the quantum energy transport model of describing the dynamics of an open quantum system of N-levels is presented. We deduce the structure of the invariant states which can be recovered by transporting states supported on the first level.

The similarity principle is an extension of the principle of thermal relaxation that naturally arises in the stochastic limit of quantum theory. We construct examples of Low Density Limit (LDL) generators, associated to an environment state in equilibrium at inverse temperature β, which admit non-(β, H S )-equilibrium states. We prove that in some cases, the attraction domain of the (β, H S )-equilibrium state is empty. This means that the similarity principle, in its original thermodynamical formulation, can be broken in the LDL limit. This result is obtained as a consequence of a more general phenomenon: the role of degeneracies in the spectrum of the Liouvillian of the system Hamiltonian associated to the generator. We start from the definition of LDL type generators given in [5] and we introduce a finer classification of these generators based on the above degeneracies. The simplest subclass, called 2-generic, is a nontrivial extension of the generators associated to the so-called Λ and V configurations, widely used in quantum optics and involving 2 levels of the system Hamiltonian. Since each 2-generic block involves 3 or 4 levels of the system Hamiltonian we expect that they can reveal some interesting new physical phenomenon, as it happened in the 2-level case. In the last section, we restrict our attention to a 3-level system with a Hamiltonian that is associated to a class of 2-generic LDL generators. Finally, we prove that, for some LDL generators in this class the statement formulated at the beginning holds true.

We study a family of quantum Markov semigroups with circulant structure. We obtain a complete description of the spectral representation for the Lindbladian and, putting it together with some purely probabilistic properties of a classical associated process, we can study asymptotic properties, invariant states, quantum-detailed balance conditions and reducibility. In particular, in the reducible case, we can construct a generator on a lower-dimensional space which can fully describe the original circulant semigroup.

We characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purely atomic algebra and certain operator subspaces associated with it in a natural way. From this result, we also establish a characterization of generators of quantum Markov semigroups of weak coupling limit type associated with a nondegenerate Hamiltonian.

We characterize the stationary states of an excitation energy transfer model in quantum many-particle systems [Y. Aref’eva, I. Volovich and S. Kozyrev, Stochastic limit method and interference in quantum many-particles systems, Theor. Math. Phys. 183(3) (2015) 782–799] as well as the stationary states of a quantum photosynthesis model [S. Kozyrev and I. Volovich, Dark states in quantum photosynthesis, arXiv:1603.07182v1 [physics.bio-ph]] in terms of a transport operator. It turns out that, apart from the ground state, all invariant states of the excitation energy transport model are entangled. For the photosynthesis model, any invariant state in the commutant of the system Hamiltonian is a mixed bright–dark state in the sense of [S. Kozyrev and I. Volovich, Dark states in quantum photosynthesis, arXiv:1603.07182v1 [physics.bio-ph]] and it is pure dark if and only if the bright vector belongs to the kernel of this state.

We introduce three new principles: the nonlinear Boltzmann–Gibbs prescription, the local KMS condition and the generalized detailed balance (GDB) condition. We prove the equivalence of the first two under general conditions and we discuss a master equation formulation of the third one.

Motivated by applications in a wide variety of fields, we study the communication (or congruence) classes of a continuous time Markov walk on a circulant graph. This study is connected to some periodicity properties of the spectrum of the infinitesimal generator and has immediate consequences on the asymptotic behavior of the process. Some sharp bounds for the spectral gap are deduced.

We study detailed balance and non-equilibrium steady states of a Markov generator of weak coupling limit type, modeling absorption and simultaneous emission of [Formula: see text]- and [Formula: see text]-photons, with [Formula: see text]. In the case [Formula: see text], under natural constraints on the absorption and emission rates, there exist infinitely many non-equilibrium steady states which are convex linear combination of even and odd states.

Quantum Markov semigroups (QMSs), also called quantum dynamical semigroups in the physics literature, were introduced in the seventies by a number of physicists to model the non-unitary evolution of quantum systems interacting with the external environment. From the mathematical point of view, they are semigroups of positive (or completely positive) maps on an operator algebra, continuous in the time variable with respect to some natural topology. They have been also studied as mathematical generalization of classical Markov semigroups, namely semigroups on commutative algebras of continuous functions. Since its introduction, the notion of a quantum Markov semigroup has been intensively studied by both physicists and mathematicians. Nowadays they play a fundamental role in models of open quantum systems, non-equilibrium phenomena (entropy methods), decoherence, quantization and orthogonal polynomials, (interacting) Fock spaces and other related topics. Problems on QMSs often constitute a difficult mathematical challenge because several methods developed for classical Markov semigroups (e.g. conditioning, coupling) do not work in this framework and the development of new ones is now beginning. The main outcomes of these investigations will be a deeper understanding of the underlying non-commutative mathematics and new results on mathematical models for open quantum systems. The workshop gathered researches from many countries in the world working on this research topic, at the crossroad between functional analysis, quantum mechanics, probability and operator theory, with different points of views and goals but very close techniques.

We introduce the notion of Θ-KMS adjoint of a quantum Markov semigroup, which is identified with the time reversed semigroup. The break of Θ-KMS symmetry, or Θ-standard quantum detailed balance in the sense of Fagnola–Umanità,11 is measured by means of the von Neumann relative entropy of states associated with the semigroup and its Θ-KMS adjoint.

We give an estimate for the off-diagonal (quantum) gap for the n-photon absorption-emission process.