Invariant States for a Quantum Markov Semigroup Constructed from Quantum Bernoulli Noises

2018 ◽  
Vol 25 (04) ◽  
pp. 1850019
Author(s):  
Jinshu Chen
Keyword(s):  
Commutation Relation ◽  
Detailed Balance ◽  
Markov Semigroup ◽  
Equal Time ◽  
Detailed Balance Condition ◽  
Balance Condition ◽  
The Family ◽  
Invariant States ◽  
Quantum Markov Semigroup ◽  
Creation Operators

Quantum Bernoulli noises are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal-time. In this paper, we consider a quantum Markov semigroup constructed from quantum Bernoulli noises. Among others, we show that the semigroup has infinitely many faithful invariant states that are diagonal, and satisfies the quantum detailed balance condition.

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).

Some non-equilibrium invariant states for the asymmetric exclusion QMS at level one

2015 ◽  
Vol 18 (01) ◽  
pp. 1550007 ◽  
Author(s):  
Julio C. García ◽  
Fernando Guerrero-Poblete
Keyword(s):  
Detailed Balance ◽  
Necessary Condition ◽  
Invariant State ◽  
Detailed Balance Condition ◽  
Balance Condition ◽  
Starting Point ◽  
Invariant States ◽  
Asymmetric Exclusion ◽  
The One ◽  
Non Equilibrium

We review the Asymmetric Exclusion QMS in the light of new results, taking as a starting point the dynamics in the one particle space. We give a condition for the Asymmetric Exclusion QMS to be conservative, prove that an invariant state is necessarily diagonal and give conditions on eigenvalues of such an invariant state. We also give, a necessary condition to annul the generator of the predual semigroup; with this and the weighted detailed balance condition, we propose a method to construct some non-equilibrium invariant states.

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.

scholarly journals Revisit on the thermodynamic stability of Hořava–Lifshitz black hole

2018 ◽  
Vol 27 (04) ◽  
pp. 1850048
Author(s):  
Xudong Meng ◽  
Ruihong Wang
Keyword(s):  
Black Hole ◽  
Thermodynamic Properties ◽  
Thermodynamic Stability ◽  
Detailed Balance ◽  
De Sitter ◽  
Thermodynamic Variable ◽  
Detailed Balance Condition ◽  
First Law Of Thermodynamics ◽  
Balance Condition ◽  
Lifshitz Black Hole

We study the thermodynamic properties of the black hole derived in Hořava–Lifshitz (HL) gravity without the detailed-balance condition. The parameter [Formula: see text] in the HL black hole plays the same role as that of the electric charge in the Reissner–Nordström-anti-de Sitter (RN-AdS) black hole. By analogy, we treat the parameter [Formula: see text] as the thermodynamic variable and obtain the first law of thermodynamics for the HL black hole. Although the HL black hole and the RN-AdS black hole have the similar mass and temperature, due to their very different entropy, the two black holes have very different thermodynamic properties. By calculating the heat capacity and the free energy, we analyze the thermodynamic stability of the HL black hole.

TWO-PHOTON ABSORPTION AND EMISSION PROCESS

2005 ◽  
Vol 08 (04) ◽  
pp. 573-591 ◽  
Author(s):  
FRANCO FAGNOLA ◽  
ROBERTO QUEZADA
Keyword(s):  
Detailed Balance ◽  
Photon Absorption ◽  
Positive Temperature ◽  
Stationary States ◽  
Emission Process ◽  
Absorption And Emission ◽  
Detailed Balance Condition ◽  
Two Photon Absorption ◽  
Two Photon ◽  
Invariant States

We analyze the two-photon absorption and emission process and characterize the stationary states at zero and positive temperature. We show that entangled stationary states exist only at zero temperature and, at positive temperature, there exists infinitely many commuting invariant states satisfying the detailed balance condition.

Quantum Feller semigroup in terms of quantum Bernoulli noises

2020 ◽  
pp. 2150015
Author(s):  
Jinshu Chen
Keyword(s):  
Local Structure ◽  
Identity Operator ◽  
Sufficient Condition ◽  
Markov Semigroups ◽  
Feller Semigroup ◽  
Abelian Subalgebra ◽  
Feller Property ◽  
The Family ◽  
Invariant States ◽  
Creation Operators

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to construct a class of quantum Markov semigroups from QBN which enjoy Feller property. As an application of our results, we examine a special quantum Feller semigroup associated with QBN, when it reduced to a certain Abelian subalgebra, the semigroup gives rise to the semigroup generated by Ornstein–Uhlenbeck operator. Finally, we find a sufficient condition for the existence of faithful invariant states that are diagonal for the semigroup.

Convergence of Markov chains in the relative supremum norm

2000 ◽  
Vol 37 (4) ◽  
pp. 1074-1083 ◽  
Author(s):  
Lars Holden
Keyword(s):  
Markov Chain ◽  
Detailed Balance ◽  
Continuity Condition ◽  
Supremum Norm ◽  
Weak Continuity ◽  
Norm Convergence ◽  
Detailed Balance Condition ◽  
Balance Condition ◽  
Qualitative Understanding ◽  
Doeblin Condition

It is proved that the strong Doeblin condition (i.e., ps(x,y) ≥ asπ(y) for all x,y in the state space) implies convergence in the relative supremum norm for a general Markov chain. The convergence is geometric with rate (1 - as)1/s. If the detailed balance condition and a weak continuity condition are satisfied, then the strong Doeblin condition is equivalent to convergence in the relative supremum norm. Convergence in other norms under weaker assumptions is proved. The results give qualitative understanding of the convergence.

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