Generators of KMS Symmetric Markov Semigroups on $${\mathcal{B}({\rm h})}$$ Symmetry and Quantum Detailed Balance

Fagnola, Franco; Umanità, Veronica

doi:10.1007/s00220-010-1011-1

Cited by 55 publications

(67 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Already in these papers, one finds several different, and nonequivalent, definitions. A number of more recent investigations [1,24,26,47,69] have added to the variety of meanings attached to this term. We therefore carefully explain the context of the definition that we use here, and why it is the most natural for our purposes.…”

Section: Detailed Balancementioning

confidence: 99%

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

Cárlen

Maas

2017

Journal of Functional Analysis

133

250

View full text Add to dashboard Cite

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.

show abstract

Section: Detailed Balancementioning

confidence: 99%

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

Cárlen

Maas

2017

Journal of Functional Analysis

133

250

View full text Add to dashboard Cite

show abstract

“…In the case where there is a faithful invariant state, it is not difficult to show that the quantum detailed balance condition (in most of its quantum formulations as in [17]) holds.…”

Section: Terms In the Above Summations Vanish Becausementioning

confidence: 99%

Quadratic open quantum harmonic oscillator

2020

Self Cite

View full text Add to dashboard Cite

We study the quantum open system evolution described by a Gorini-Kossakowski-Sudarshan-Lindblad generator with creation and annihilation operators arising in Fock representations of the sl 2 Lie algebra. We show that any initial density matrix evolves to a fully supported density matrix and converges towards a unique equilibrium state. We show that the convergence is exponentially fast and we exactly compute the rate for a wide range of parameters. We also discuss the connection with the two-photon absorption and emission process.

show abstract

“…More results on the quantum detailed balance can be found in e.g. [37,[48][49][50] and the references therein.…”

Section: B Quantum Markov Semigroup With Gns-detailed Balancementioning

confidence: 99%

Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance

Cao

2019

Journal of Mathematical Physics

View full text Add to dashboard Cite

We study the entropy production of the sandwiched Rényi divergence under the primitive Lindblad equation with GNS-detailed balance. We prove that the Lindblad equation can be identified as the gradient flow of the sandwiched Rényi divergence of any order α ∈ (0, ∞). This extends a previous result by Carlen and Maas [Journal of Functional Analysis, 273(5), 1810-1869] for the quantum relative entropy (i.e., α = 1). Moreover, we show that the sandwiched Rényi divergence of any order α ∈ (0, ∞) decays exponentially fast under the time-evolution of such a Lindblad equation.

show abstract

Generators of KMS Symmetric Markov Semigroups on $${\mathcal{B}({\rm h})}$$ Symmetry and Quantum Detailed Balance

Cited by 55 publications

References 26 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

Quadratic open quantum harmonic oscillator

Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance

Contact Info

Product

Resources

About