QUANTUM MARKOV SEMIGROUPS: PRODUCT SYSTEMS AND SUBORDINATION

We show that if a product system comes from a quantum Markov semigroup, then it carries a natural Borel structure with respect to which the semigroup may be realized in terms of a measurable representation. We show, too, that the dual product system of a Borel product system also carries a natural Borel structure. We apply our analysis to study the order interval consisting of all quantum Markov semigroups that are subordinate to a given one.

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Structure of generic quantum Markov semigroup

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025717500126 ◽

2017 ◽

Vol 20 (02) ◽

pp. 1750012 ◽

Cited By ~ 6

Author(s):

R. Carbone ◽

E. Sasso ◽

V. Umanità

Keyword(s):

Asymptotic Behavior ◽

Markov Process ◽

Fixed Points ◽

Free Algebra ◽

Markov Semigroup ◽

Markov Semigroups ◽

Diagonal Part ◽

Invariant States ◽

Quantum Markov Semigroup

In this paper, we study some relevant properties of generic quantum Markov semigroups, in particular related to their asymptotic behavior. We can describe the structure of the set of fixed points and of the invariant states in terms of the Hamiltonian’s spectrum and of the communication classes of the classical Markov process associated with the diagonal part of the semigroup. Moreover we study the decoherence-free algebra and we complete the characterization of environmental decoherence for a generic quantum Markov semigroup.

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Structure of block quantum dynamical semigroups and their product systems

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025720500010 ◽

2020 ◽

Vol 23 (01) ◽

pp. 2050001

Author(s):

B. V. Rajarama Bhat ◽

U. Vijaya Kumar

Keyword(s):

Von Neumann Algebra ◽

Markov Semigroup ◽

Map Building ◽

Quantum Dynamical Semigroup ◽

Von Neumann ◽

Quantum Dynamical ◽

Product Systems ◽

Dynamical Semigroup ◽

Quantum Markov Semigroup ◽

Unit Formula

Paschke’s version of Stinespring’s theorem associates a Hilbert [Formula: see text]-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a [Formula: see text]-algebra [Formula: see text] one may associate an inclusion system [Formula: see text] of Hilbert [Formula: see text]-[Formula: see text]-modules with a generating unit [Formula: see text]. Suppose [Formula: see text] is a von Neumann algebra, consider [Formula: see text], the von Neumann algebra of [Formula: see text] matrices with entries from [Formula: see text]. Suppose [Formula: see text] with [Formula: see text] is a QDS on [Formula: see text] which acts block-wise and let [Formula: see text] be the inclusion system associated to the diagonal QDS [Formula: see text] with the generating unit [Formula: see text] It is shown that there is a contractive (bilinear) morphism [Formula: see text] from [Formula: see text] to [Formula: see text] such that [Formula: see text] for all [Formula: see text] We also prove that any contractive morphism between inclusion systems of von Neumann [Formula: see text]-[Formula: see text]-modules can be lifted as a morphism between the product systems generated by them. We observe that the [Formula: see text]-dilation of a block quantum Markov semigroup (QMS) on a unital [Formula: see text]-algebra is again a semigroup of block maps.

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Gaussian Quantum Markov Semigroups on a One-Mode Fock Space: Irreducibility and Normal Invariant States

Open Systems & Information Dynamics ◽

10.1142/s1230161221500013 ◽

2021 ◽

Vol 28 (01) ◽

pp. 2150001

Author(s):

J. Agredo ◽

F. Fagnola ◽

D. Poletti

Keyword(s):

Existence And Uniqueness ◽

Fock Space ◽

Sufficient Conditions ◽

Quantum Oscillator ◽

Markov Semigroup ◽

Markov Semigroups ◽

Type Condition ◽

Invariant States ◽

Quantum Markov Semigroup ◽

Necessary And Sufficient

We consider the most general Gaussian quantum Markov semigroup on a one-mode Fock space, discuss its construction from the generalized GKSL representation of the generator. We prove the known explicit formula on Weyl operators, characterize irreducibility and its equivalence to a Hörmander type condition on commutators and establish necessary and sufficient conditions for existence and uniqueness of normal invariant states. We illustrate these results by applications to the open quantum oscillator and the quantum Fokker-Planck model.

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THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP WITH UNBOUNDED GENERATOR

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025710004176 ◽

2010 ◽

Vol 13 (03) ◽

pp. 413-433 ◽

Cited By ~ 23

Author(s):

AMEUR DHAHRI ◽

FRANCO FAGNOLA ◽

ROLANDO REBOLLEDO

Keyword(s):

Steady State ◽

Regularity Conditions ◽

Markov Semigroup ◽

Invariant State ◽

Markov Semigroups ◽

Completely Positive Maps ◽

Completely Positive ◽

Infinite Dimensional ◽

Positive Maps ◽

Quantum Markov Semigroup

Let [Formula: see text] be a quantum Markov semigroup on [Formula: see text] with a faithful normal invariant state ρ. The decoherence-free subalgebra [Formula: see text] of [Formula: see text] is the biggest subalgebra of [Formula: see text] where the completely positive maps [Formula: see text] act as homomorphisms. When [Formula: see text] is the minimal semigroup whose generator is represented in a generalised GKSL form [Formula: see text], with possibly unbounded H, Lℓ, we show that [Formula: see text] coincides with the generalised commutator of [Formula: see text] under some natural regularity conditions. As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and Lℓ. We give examples of quantum Markov semigroups [Formula: see text], with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.

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The Θ-KMS adjoint and time reversed quantum Markov semigroups

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025715500162 ◽

2015 ◽

Vol 18 (03) ◽

pp. 1550016 ◽

Cited By ~ 5

Author(s):

Jorge R. Bolaños-Servin ◽

Roberto Quezada

Keyword(s):

Relative Entropy ◽

Detailed Balance ◽

Markov Semigroup ◽

Markov Semigroups ◽

Standard Quantum ◽

Von Neumann ◽

Quantum Markov Semigroup

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.

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Quantum Markov Semigroups with Unbounded Generator and Time Evolution of the Support Projection of a State

Open Systems & Information Dynamics ◽

10.1142/s123016121850004x ◽

2018 ◽

Vol 25 (01) ◽

pp. 1850004

Author(s):

Souhir Gliouez ◽

Skander Hachicha ◽

Ikbel Nasroui

Keyword(s):

Time Evolution ◽

Markov Semigroup ◽

Markov Semigroups ◽

Quantum Version ◽

Quantum Markov Semigroup ◽

Support Projection

We characterize the support projection of a state evolving under the action of a quantum Markov semigroup with unbounded generator represented in the generalized GKSL form and a quantum version of the classical Lévy-Austin-Ornstein theorem.

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THE ASYMMETRIC EXCLUSION QUANTUM MARKOV SEMIGROUP

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025709003781 ◽

2009 ◽

Vol 12 (03) ◽

pp. 367-385 ◽

Cited By ~ 5

Author(s):

LEOPOLDO PANTALEÓN-MARTÍNEZ ◽

ROBERTO QUEZADA

Keyword(s):

Detailed Balance ◽

Markov Semigroup ◽

Markov Semigroups ◽

Quantum Processes ◽

Detailed Balance Condition ◽

Index Site ◽

Classical Invariant ◽

Invariant States ◽

Quantum Markov Semigroup ◽

Asymmetric Exclusion

In this paper we study a class of quantum Markov semigroups whose restriction to an abelian sub-algebra coincides, on the configurations with finite support, with the exclusion type semigroups introduced in Liggett's book14 of exchange rates [Formula: see text] not symmetric in the index site r, s. We find a sufficient condition for the existence of infinitely many faithful diagonal (or classical) invariant states for the semigroup, that satisfy a quantum detailed balance condition. This class of semigroups arises naturally in the stochastic limit of quantum interacting particles in the sense of Accardi and Kozyrev.1 We call these semigroups asymmetric exclusion quantum Markov semigroups and the associated processes asymmetric exclusion quantum processes.

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Generic Quantum Markov Semigroups with Degenerate Ground States: The Fock Case

Open Systems & Information Dynamics ◽

10.1142/s1230161218500105 ◽

2018 ◽

Vol 25 (02) ◽

pp. 1850010 ◽

Cited By ~ 1

Author(s):

Skander Hachicha ◽

Ikbel Nasraoui

Keyword(s):

Weak Coupling ◽

Weak Coupling Limit ◽

Markov Semigroup ◽

Zero Temperature ◽

Invariant State ◽

Markov Semigroups ◽

Initial State ◽

Arbitrary Initial State ◽

Invariant States ◽

Quantum Markov Semigroup

We consider quantum Markov semigroups arising from the weak coupling limit of a system with generic Hamiltonian coupled to a boson Fock zero temperature reservoir. We find all the invariant states of a generic quantum Markov semigroup and compute explicitly the limit invariant state explicitly starting from an arbitrary initial state. We also show that convergence is exponentially fast under some natural assumptions.

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On the range of the generator of a quantum Markov semigroup

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025715500277 ◽

2015 ◽

Vol 18 (04) ◽

pp. 1550027 ◽

Cited By ~ 1

Author(s):

Jorge R. Bolaños-Servin ◽

Franco Fagnola

Keyword(s):

Fixed Point ◽

Infinitesimal Generator ◽

Markov Semigroup ◽

Invariant State ◽

Markov Semigroups ◽

Quantum Markov Semigroup ◽

Kms Condition ◽

Fixed Point Algebra

We show that the commutant of the range of the infinitesimal generator of a norm-continuous quantum Markov semigroup on [Formula: see text], not consisting of identity maps, with a faithful normal invariant state is trivial whenever the fixed point algebra is atomic. As a consequence, two formulations of the irreversible [Formula: see text]-KMS condition proposed in Ref. 2 are equivalent for this class of quantum Markov semigroups.

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Equivalence between logarithmic Sobolev inequality and hypercontractivity in a probability gage space

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091520000383 ◽

2020 ◽

pp. 1-13

Author(s):

Zhang Lunchuan

Keyword(s):

Dirichlet Form ◽

Sobolev Inequality ◽

Logarithmic Sobolev Inequality ◽

Markov Semigroup ◽

Quantum Markov Semigroup

Abstract In this paper, we prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of a class of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space.

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