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.

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.

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.

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.

scholarly journals Bounded solutions of a difference equation with finite number of jumps of operator coefficient

2020 ◽  
Vol 12 (1) ◽  
pp. 165-172
Author(s):  
A. Chaikovs'kyi ◽  
O. Lagoda
Keyword(s):  
Difference Equation ◽  
Finite Number ◽  
Existence And Uniqueness ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Operator Coefficient ◽  
Bounded Solutions ◽  
Piecewise Constant ◽  
Variable Operator ◽  
Necessary And Sufficient

We study the problem of existence of a unique bounded solution of a difference equation with variable operator coefficient in a Banach space. There is well known theory of such equations with constant coefficient. In that case the problem is solved in terms of spectrum of the operator coefficient. For the case of variable operator coefficient correspondent conditions are known too. But it is too hard to check the conditions for particular equations. So, it is very important to give an answer for the problem for those particular cases of variable coefficient, when correspondent conditions are easy to check. One of such cases is the case of piecewise constant operator coefficient. There are well known sufficient conditions of existence and uniqueness of bounded solution for the case of one jump. In this work, we generalize these results for the case of finite number of jumps of operator coefficient. Moreover, under additional assumption we obtained necessary and sufficient conditions of existence and uniqueness of bounded solution.

scholarly journals Diagonal unitary and orthogonal symmetries in quantum theory

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 519
Author(s):  
Satvik Singh ◽  
Ion Nechita
Keyword(s):  
Sufficient Conditions ◽  
Quantum Channels ◽  
Completely Positive ◽  
Matrix Algebras ◽  
Completely Positive Matrices ◽  
Linear Maps ◽  
Positive Matrices ◽  
Invariant States ◽  
Class Of Matrices ◽  
Necessary And Sufficient

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

scholarly journals Fixed point theory in the setting of $(\alpha,\beta,\psi,\phi)$-interpolative contractions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Erdal Karapınar ◽  
Andreea Fulga ◽  
Antonio Francisco Roldán López de Hierro
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Necessary And Sufficient Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

AbstractIn this manuscript we introduce the notion of $(\alpha,\beta,\psi,\phi)$ ( α , β , ψ , ϕ ) -interpolative contraction that unifies and generalizes significant concepts: Proinov type contractions, interpolative contractions, and ample spectrum contraction. We investigate the necessary and sufficient conditions to guarantee existence and uniqueness of the fixed point of such mappings.

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.

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