Spectral Analysis and Feller Property for Quantum Ornstein-Uhlenbeck Semigroups

Cipriani, Fabio; Fagnola, Franco; Lindsay, J. Martin

doi:10.1007/s002200050773

Cited by 42 publications

(41 citation statements)

References 14 publications

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“…The operator L β is in fact the generator of an ergodic QMS, as shown in [17]. These authors construct the QMS first on the infinite-dimensional analog of H A , which in this case strictly contains A, and then show that the resulting semigroup has the Feller property; i.e., it preserves A.…”

Section: The Bose Ornstein-uhlenbeck Semigroupmentioning

confidence: 99%

“…These authors construct the QMS first on the infinite-dimensional analog of H A , which in this case strictly contains A, and then show that the resulting semigroup has the Feller property; i.e., it preserves A. In [17], another detailed balance condition based on self-adjointness with respect to the KMS inner product is used. However, Theorem 2.9 and what we have said above about the modular operator shows that the semigroup also satisfies the σ β -DBC as defined here.…”

Section: The Bose Ornstein-uhlenbeck Semigroupmentioning

confidence: 99%

See 1 more Smart Citation

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

Cárlen

Maas

2017

Journal of Functional Analysis

114

250

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Abstract. We study a class of ergodic quantum Markov semigroups on finitedimensional unital C * -algebras. These semigroups have a unique stationary state σ, and we are concerned with those that satisfy a quantum detailed balance condition with respect to σ. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to σ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the 2-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical 2-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.

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Section: The Bose Ornstein-uhlenbeck Semigroupmentioning

confidence: 99%

Section: The Bose Ornstein-uhlenbeck Semigroupmentioning

confidence: 99%

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

Cárlen

Maas

2017

Journal of Functional Analysis

114

250

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“…For given y ∈ M 1/2 and Q = Q * ∈ M 1/2 , we first consider the following Lindblad type generator of a q.d.s. on M : 11) and define the operator H on H by…”

Section: Notation Definitions and Main Resultsmentioning

confidence: 99%

“…One of the reasons is that the general structure of Dirichlet forms for non-tracial states is not well-understood compared to the tracial case [2,3,6,12]. For constructions of Dirichlet forms for non-tracial states, we refer to [8,9,11,18,20,21,25,23] and the references there in. In [23], we gave a general construction method of Dirichlet forms on standard forms of von Neumann algebras.…”

Section: Introductionmentioning

confidence: 99%

Remarks on the Structure of Dirichlet Forms on Standard Forms of Von Neumann Algebras

Park

2005

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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For a von Neumann algebra M acting on a Hilbert space H with a cyclic and separating vector ξ 0 , we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (M, ξ 0 ). For a general Lindblad type generator L of a conservative quantum dynamical semigroup on M, we give sufficient conditions so that the operator H induced by L via the symmetric embedding of M into H to be self-adjoint. It turns out that the self-adjoint operator H can be written in the form of a Dirichlet operator associated to a Dirichlet form given in [23]. In order to make the connection possible, we also extend the range of applications of the formula in [23].

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“…Indeed, all Markov semigroups arising from the stochastic limit [ALV01] do. Some of them like the so-called quantum Ornstein-Uhlenbeck semigroup admit an infinite number of such invariant subalgebras; see Cipriani, Fagnola and Lindsay [CFL00].…”

mentioning

confidence: 99%