Non-commutative symmetric Markov semigroups

Davies, E. B.; Lindsay, J. Martin

doi:10.1007/bf02571804

Cited by 73 publications

(62 citation statements)

References 23 publications

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“…A powerful non-commutative extension of this theory has been developed starting with the early work of Gross [31,32], and continuing with [3,15,16,18,21,29]. We shall not need the whole theory at present, but the Dirichlet form representation of the generator of a QMS will be useful to us.…”

Section: Dirichlet Form Representation Associated To a Quantum Markovmentioning

confidence: 99%

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

Cárlen

Maas

2017

Journal of Functional Analysis

126

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: Dirichlet Form Representation Associated To a Quantum Markovmentioning

confidence: 99%

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

Cárlen

Maas

2017

Journal of Functional Analysis

126

250

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“…In section 2 we establish the correspondence between Dirichlet forms, sub-Markovian semigroups and sub-Markovian resolvents, thus generalizing the results of S. Albeverio and R. Høegh-Krohn ( [AH], see also [DL1]) to the non symmetric case. This is made adapting the non-symmetric abelian definitions and results in [MR] to the non commutative (semifinite) case.…”

Section: Introductionmentioning

confidence: 78%

“…The theory of non commutative Dirichlet forms, which originated from the pioneering examples of L. Gross [G] and the general analysis of S. Albeverio and R. Høegh-Krohn [AH] (see also [AHO]), has nowadays drawn a renewed interest between researchers ( [DL1], [DL2], [DR], [D3], [Sa], [GL] and [Ci]). There are different reasons which, in our opinion, explain (and justify) the recent activity in this area.…”

Section: Introductionmentioning

confidence: 99%

Non-symmetric Dirichlet Forms on Semifinite von Neumann Algebras

Guido¹,

Isola²,

Scarlatti³

1996

Journal of Functional Analysis

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“…. , n, let us introduce a 2n-dimensional column vector of observables 9) where the superscript T denotes a transpose. Then the canonical commutation relations (CCR) can be compactly expressed as…”

Section: The Algebra Of Canonical Commutation Relations (Ccr)mentioning

confidence: 99%

Contractivity properties of a quantum diffusion semigroup

Datta

Pautrat

Rouzé

2017

Journal of Mathematical Physics

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We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of a state with positive Wigner function, evolving under the action of the semigroup, decrease at least inverse polynomially in time, while its entropy increases at least logarithmically in time.

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Non-commutative symmetric Markov semigroups

Cited by 73 publications

References 23 publications

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

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

Non-symmetric Dirichlet Forms on Semifinite von Neumann Algebras

Contractivity properties of a quantum diffusion semigroup

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