On the range of the generator of a quantum Markov semigroup

2015 ◽  
Vol 18 (04) ◽  
pp. 1550027 ◽  
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.

ALGEBRAIC CONDITIONS FOR CONVERGENCE OF A QUANTUM MARKOV SEMIGROUP TO A STEADY STATE

2008 ◽  
Vol 11 (03) ◽  
pp. 467-474 ◽  
Author(s):  
FRANCO FAGNOLA ◽  
ROLANDO REBOLLEDO
Keyword(s):  
Fixed Point ◽  
Steady State ◽  
Spin Chain ◽  
Chain Model ◽  
Markov Semigroup ◽  
Invariant State ◽  
Spin Chain Model ◽  
Quantum Markov Semigroup ◽  
Fixed Point Algebra ◽  
Uniformly Continuous

Let [Formula: see text] be a uniformly continuous quantum Markov semigroup on [Formula: see text] with generator represented in a standard GKSL form [Formula: see text] and a faithful normal invariant state ρ. In this note we give new algebraic conditions for proving that [Formula: see text] converges towards a steady state, possibly different from ρ. Indeed, we show that this happens whenever the commutator of [Formula: see text] (i.e. its fixed point algebra) coincides with the commutator of [Formula: see text] (where δH(X) = [H, X]) for some n ≥ 1. As an application we discuss the convergence to the unique invariant state of a spin chain model.

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

2010 ◽  
Vol 13 (03) ◽  
pp. 413-433 ◽  
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.

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

2018 ◽  
Vol 25 (02) ◽  
pp. 1850010 ◽  
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.

Structure of generic quantum Markov semigroup

2017 ◽  
Vol 20 (02) ◽  
pp. 1750012 ◽  
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.

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

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.

The Θ-KMS adjoint and time reversed quantum Markov semigroups

2015 ◽  
Vol 18 (03) ◽  
pp. 1550016 ◽  
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.

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

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.

scholarly journals QUANTUM MARKOV SEMIGROUPS: PRODUCT SYSTEMS AND SUBORDINATION

2007 ◽  
Vol 18 (06) ◽  
pp. 633-669 ◽  
Author(s):  
PAUL S. MUHLY ◽  
BARUCH SOLEL
Keyword(s):  
Markov Semigroup ◽  
Order Interval ◽  
Markov Semigroups ◽  
Product System ◽  
Product Systems ◽  
Borel Structure ◽  
Quantum Markov Semigroup

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.

THE ASYMMETRIC EXCLUSION QUANTUM MARKOV SEMIGROUP

2009 ◽  
Vol 12 (03) ◽  
pp. 367-385 ◽  
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.

scholarly journals GENERATORS OF DETAILED BALANCE QUANTUM MARKOV SEMIGROUPS

2007 ◽  
Vol 10 (03) ◽  
pp. 335-363 ◽  
Author(s):  
FRANCO FAGNOLA ◽  
VERONICA UMANITÀ
Keyword(s):  
Detailed Balance ◽  
Representation Formula ◽  
Markov Semigroup ◽  
Invariant State ◽  
Open Problems ◽  
Detailed Balance Condition ◽  
Balance Condition ◽  
Adjoint Semigroup ◽  
Quantum Markov Semigroup ◽  
Scalar Products

For a quantum Markov semigroup [Formula: see text] on the algebra [Formula: see text] with a faithful invariant state ρ, we can define an adjoint [Formula: see text] with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterizing adjoints [Formula: see text] that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H, Lk in the Gorini–Kossakowski–Sudarshan–Lindblad representation [Formula: see text] of the generator of [Formula: see text]. We study the adjoint semigroup with respect to both scalar products 〈a, b〉 = tr (ρa*b) and 〈a, b〉 = tr (ρ1/2a*ρ1/2 b).

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