THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP ON $\mathcal{B}({\rm{h}})$

2011 ◽  
Author(s):  
A. DHAHRI ◽  
F. FAGNOLA ◽  
R. REBOLLEDO
Keyword(s):  
Markov Semigroup ◽  
Quantum Markov Semigroup

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

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.

Support projection of state and a quantum Lévy–Austin–Ornstein theorem

2014 ◽  
Vol 17 (03) ◽  
pp. 1450020 ◽  
Author(s):  
Skander Hachicha
Keyword(s):  
Open Quantum Systems ◽  
Quantum Systems ◽  
Markov Semigroup ◽  
Open Quantum ◽  
Quantum Analogue ◽  
Quantum Markov Semigroup ◽  
Support Projection

We prove two characterizations of the support projection of a state evolving under the action of a quantum Markov semigroup and a quantum analogue of the Lévy–Austin–Ornstein theorem. We discuss applications to open quantum systems.

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.

Stability of the Markov (conservativity) property under perturbations

2020 ◽  
Vol 23 (02) ◽  
pp. 2050009
Author(s):  
Dharmendra Kumar ◽  
Kalyan B. Sinha ◽  
Sachi Srivastava
Keyword(s):  
Markov Semigroup ◽  
Quantum Markov Semigroup

It is shown that if a Quantum Markov semigroup is “multiplicatively” perturbed under suitable conditions, the resulting minimal semigroup remains Markov (or conservative).

scholarly journals Structure of block quantum dynamical semigroups and their product systems

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.

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 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.

A Wasserstein-type Distance to Measure Deviation from Equilibrium of Quantum Markov Semigroups

2013 ◽  
Vol 20 (02) ◽  
pp. 1350009 ◽  
Author(s):  
Julián Agredo
Keyword(s):  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Optimal Transport ◽  
Detailed Balance ◽  
Wasserstein Distance ◽  
Markov Semigroup ◽  
Bounded Operators ◽  
Von Neumann ◽  
Detailed Balance Condition ◽  
Quantum Markov Semigroup

In this paper we define a distance W between states in the non-commutative von Neumann algebra [Formula: see text] of bounded operators on a separable Hilbert space [Formula: see text], in order to measure deviations from equilibrium using a rate ep W(·). The restriction of W to the diagonal subalgebra of [Formula: see text] coincides with the Wasserstein distance used in optimal transport. Moreover, if ρ is a normal invariant state of a quantum Markov semigroup [Formula: see text], then ep W(ρ) = 0 if and only if a detailed balance condition holds.

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.

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