Quantum Markov Semigroups Research Articles

A note on invariant states of Gaussian quantum Markov semigroups

Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is characterized either by the property of their predual that preserves quantum Gaussian states. In this paper, show that gaussian QMSs can have only Gaussian invariant states if the spectrum of the drift matrix does not contain points on the imaginary axis. Moreover, in this case, any initial state converges in trace norm toward a unique gaussian invariant state.

Local Detailed Balance Invariant States of Markov Generators of Low Density Limit

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.

Christensen–Evans theorem and extensions of GNS-symmetric quantum Markov semigroups

In this article we prove a refined version of the Christensen–Evans theorem for generators of uniformly continuous GNS-symmetric quantum Markov semigroups. We use this result to show the existence of GNS-symmetric extensions of GNS-symmetric quantum Markov semigroups. In particular, this implies that the generators of GNS-symmetric quantum Markov semigroups on finite-dimensional von Neumann algebra can be written in the form specified by Alicki's theorem.

On irreducibility and positivity improvement of norm continuous quantum Markov semigroups

We show that for a norm continuous quantum Markov semigroup irreducibility and positivity improvement for any time [Formula: see text] are equivalent under some general conditions that hold, in particular, in the finite-dimensional case. This result is applied to quantum Markov semigroups of weak coupling limit type. We also discuss the discrete-time case.

The Fast Recurrent Subspace on an N-Level Quantum Energy Transport Model

The fast recurrent subspace (the biggest support of all invariant states) of a weak coupling limit type quantum Markov semigroup modeling a quantum transport open system of [Formula: see text]-energy levels is determined. This is achieved by characterizing the structure of all the invariant states and their spectra in terms of a natural generalization of the discrete Fourier transform operator. Finally, the attraction domains and long-time behaviour of the evolution are studied on hereditary subalgebras where faithful invariant states exist.

Trace distance ergodicity for quantum Markov semigroups

We discuss the quantitative ergodicity of quantum Markov semigroups in terms of the trace distance from the stationary state, providing a general criterion based on the spectral decomposition of the Lindblad generator. We then apply this criterion to the bosonic and fermionic Ornstein-Uhlenbeck semigroups and to a family of quantum Markov semigroups parametrized by semisimple Lie algebras and their irreducible representations, in which the Lindblad generator is given by the adjoint action of the Casimir element.

Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

In deformation-rigidity theory, it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra \mathbb{C}[\Gamma] with \Gamma a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten \mathcal{S}_{p} class p \in [2, \infty) , then the n -fold tensor power H^{\otimes n}_{\Gamma} for n \geq \frac{p}{2} is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups, we give a number of characterizations of having coefficients in \mathcal{S}_{p} for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient- \mathcal{S}_{p} property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups, (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by Sinclair for discrete groups admitting a proper cocycle into a p -integrable representation.

Thermodynamic Formalism for Continuous-Time Quantum Markov Semigroups: the Detailed Balance Condition, Entropy, Pressure and Equilibrium Quantum Processes

Let [Formula: see text] denote the set of [Formula: see text] by [Formula: see text] complex matrices. Consider continuous time quantum semigroups [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator. If we assume that [Formula: see text], we will call [Formula: see text], [Formula: see text] a quantum Markov semigroup. Given a stationary density matrix [Formula: see text], for the quantum Markov semigroup [Formula: see text], [Formula: see text], we can define a continuous time stationary quantum Markov process, denoted by [Formula: see text], [Formula: see text] Given an a priori Laplacian operator [Formula: see text], we will present a natural concept of entropy for a class of density matrices on [Formula: see text]. Given a Hermitian operator [Formula: see text] (which plays the role of a Hamiltonian), we will study a version of the variational principle of pressure for [Formula: see text]. A density matrix [Formula: see text] maximizing pressure will be called an equilibrium density matrix. From [Formula: see text] we will derive a new infinitesimal generator [Formula: see text]. Finally, the continuous time quantum Markov process defined by the semigroup [Formula: see text], [Formula: see text], and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian [Formula: see text]. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian [Formula: see text].

Exponential convergence in the trace norm and time-inhomogeneous Markovian evolutions on ℬ(ℋ)

We consider a sequence of operators [Formula: see text] that describes the time evolution of normal states of an open quantum system. Our objective is to analyze the asymptotic behavior of [Formula: see text], where [Formula: see text] is an arbitrary initial normal state. Our focus is on specific situations in which the operators [Formula: see text] are produced by various quantum Markov semigroups of weak coupling limit-type. To tackle this problem, we introduce a new result concerning exponential convergence to equilibrium in the trace norm for quantum Markov semigroups, provided that their induced semigroups have a spectral gap. Moreover, applications to quantum annealing are discussed.

Recurrence and transience of quantum Markov semigroups constructed form quantum Bernoulli noises

Quantum exclusion semigroup constructed form quantum Bernoulli noises can be written on diagonal and off-diagonal operators space, respectively. The diagonal parts describe a classical Markov process. In this paper, we first produce a decomposition into the sum of irreducible components determined by the class states of the associated classical Markov process and prove that each irreducible component is recurrent or transient if and only if its diagonal restriction is recurrent or transient. We then give conditions for existence of invariant states and study properties of invariant states such as detailed balance, uniqueness and convergence towards as time approaches infinity.

Derivations and KMS-Symmetric Quantum Markov Semigroups

We prove that the generator of the L^2 implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a Hilbert bimodule, extending earlier results by Cipriani and Sauvageot for tracially symmetric semigroups and the second-named author for GNS-symmetric semigroups. This result hinges on the introduction of a new completely positive map on the algebra of bounded operators on the GNS Hilbert space. This transformation maps symmetric Markov operators to symmetric Markov operators and is essential to obtain the required inner product on the Hilbert bimodule.

On the existence of derivations as square roots of generators of state-symmetric quantum Markov semigroups

Cipriani and Sauvageot have shown that for any [Formula: see text]-generator [Formula: see text] of a tracially symmetric quantum Markov semigroup on a C*-algebra [Formula: see text] there exists a densely defined derivation [Formula: see text] from [Formula: see text] to a Hilbert bimodule [Formula: see text] such that [Formula: see text]. Here, we show that this construction of a derivation can in general not be generalized to quantum Markov semigroups that are symmetric with respect to a non-tracial state. In particular, we show that all derivations to Hilbert bimodules can be assumed to have a concrete form, and then we use this form to show that in the finite-dimensional case the existence of such a derivation is equivalent to the existence of a positive matrix solution of a system of linear equations. We solve this system of linear equations for concrete examples using Mathematica to complete the proof.

Quantum Markov semigroup for open quantum system interacting with quantum Bernoulli noises

Quantum Bernoulli noises (QBNs) refer to the annihilation and creation operators acting on the space [Formula: see text] of square integrable Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. In this paper, we consider the Markov evolution of an open quantum system interacting with QBNs. Let [Formula: see text] be the system space of an open quantum system interacting with QBNs. Then [Formula: see text] just describes the coupled quantum system. In the framework of [Formula: see text], we first construct a quantum Markov semigroup that respects the interactions between the system and QBNs, and then we prove that under some mild conditions the semigroup has faithful invariant states. To support our main results mentioned above, we prove several technical propositions and theorems about operators defined in [Formula: see text]. Some other results are also obtained.

The general structure of the decoherence-free subalgebra for uniformly continuous quantum Markov semigroups

By using the decomposition of the decoherence-free subalgebra N(T) in direct integrals of factors, we obtain a structure theorem for every uniformly continuous quantum Markov semigroup. Moreover, we prove that when there exists a faithful normal invariant state, N(T) has to be atomic and decoherence takes place.

G-Circulant Quantum Markov Semigroups

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.

Entropy production of quantum Markov semigroup associated with open quantum walks on the periodic graphs

Entropy production of quantum Markov semigroup associated with open quantum walks on the periodic graphs

Stability of Logarithmic Sobolev Inequalities Under a Noncommutative Change of Measure

We generalize Holley-Stroock's perturbation argument from commutative to quantum Markov semigroups. As a consequence, results on (complete) modified logarithmic Sobolev inequalities and logarithmic Sobolev inequalities for self-adjoint quantum Markov process can be used to prove estimates on the exponential convergence in relative entropy of quantum Markov systems which preserve a fixed state. This leads to estimates for the decay to equilibrium for coupled systems and to estimates for mixed state preparation times using Lindblad operators. Our techniques also apply to discrete time settings, where we show that the strong data processing inequality constant of a quantum channel can be controlled by that of a corresponding unital channel.

On irreducibility of Gaussian quantum Markov semigroups

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.

On the relationships between covariance and the decoherence-free subalgebra of a quantum Markov semigroup

In this paper, we investigate if the presence of symmetries in the evolution of an open quantum system gives information about the invariant subspaces. Unfortunately, the answer is, in general, negative, but under suitable conditions we can observe that the representation through which we describe the symmetry determines a privileged family of orthogonal projections strongly correlated with the structure of the decoherence-free sub-algebra and, as a consequence, with harmonic projections.

Curvature-Dimension Conditions for Symmetric Quantum Markov Semigroups

Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz–Schur multipliers over group algebras and generalized depolarizing semigroups.