Sufficient Conditions for Conservativity of Quantum Dynamical Semigroups

Chebotarev, A. M.; Fagnola, Franco

doi:10.1006/jfan.1993.1140

Cited by 46 publications

(72 citation statements)

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“…That there exists a unique conservative σ-weakly continuous Markov semigroup on B(H o ) with L as it's generator follows from the general criteria of Chebotarev and Fagnola [CheFa93].…”

Section: Petz's Dualitymentioning

confidence: 93%

Time Reflected Markov Processes

Accardi

Mohari

1999

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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A classical stochastic process which is Markovian for its past filtration is also Markovian for its future filtration. We show with a counterexample based on quantum liftings of a finite state classical Markov chain that this property cannot hold in the category of expected Markov processes. Using a duality theory for von Neumann algebras with weights, developed by Petz on the basis of previous results by Groh and Kümmerer, we show that a quantum version of this symmetry can be established in the category of weak Markov processes in the sense of Bhat and Parthasarathy. Here time reversal is implemented by an anti-unitary operator and a weak Markov process is time reversal invariant if and only if the associated semigroup coincides with its Petz dual. This construction allows one to extend to the quantum case, both for backward and forward processes, the Misra–Prigogine–Courbage internal time operator and to show that the two operators are intertwined by the time reversal anti-automorphism.

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“…That there exists a unique conservative σ-weakly continuous Markov semigroup on B(H o ) with L as it's generator follows from the general criteria of Chebotarev and Fagnola [CheFa93].…”

Section: Petz's Dualitymentioning

confidence: 93%

Time Reflected Markov Processes

Accardi

Mohari

1999

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…In the particular case where the potential is stationary with respect to space, see (8), equation (25) reads…”

Section: White Noise Limitmentioning

confidence: 99%

Quantum Transport and Boltzmann Operators

2006

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In this paper the transport of quantum particles in time-dependent random media is studied. In the white noise limit, a quantum model for collisions is obtained. At the level of Wigner equation, this limit is described by a linear Wigner-Boltzmann equation.Nousétudions dans cet article le transport de particules quantiques dans un milieu aléatoire dépendant du temps. Dans la limite de bruit blanc, nous obtenons un modèle quantique de collison. Au niveau deséquations de Wigner, la limite est donnée par uné equation de Wigner-Boltzmann linéaire.

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“…may not be unique and conservative [8,17]. Under suitable conditions, the above equation Also the study of conservativity conditions is important in quantum probability because they play a key role in the proof of uniqueness and unitarity of solutions of an Hudson-Parthasarathy quantum stochastic differential equation [18 -20].…”

Section: Introductionmentioning

confidence: 99%

“…Chebotarev gave necessary and sufficient conditions for conservativity [8]. Some of the conditions, however, are impossible to check practically in many interesting examples.…”

Section: Introductionmentioning

confidence: 99%