Sufficient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups

Chebotarev, A. M.; Fagnola, Franco

doi:10.1006/jfan.1997.3189

Cited by 115 publications

(109 citation statements)

References 21 publications

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“…However since any useful (easy to check) sufficient condition for proving this conservativity is not really available (even in HP case) beyond conditions really similar to those of our section 2 to get mild solutions (cf. [8]), we don't enter in this general question here. Let us conclude with two remarks.…”

Section: Introductionmentioning

confidence: 96%

A free stochastic partial differential equation

Dabrowski¹

2014

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. We get stationary solutions of a free stochastic partial differential equation.As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also the von Neumann algebra R ω embeddable. This includes an N-tuple of q-Gaussian random variables e.g. for |q|N ≤ 0.13. IntroductionIn a fundamental series of papers, Voiculescu introduced analogs of entropy and Fisher information in the context of free probability theory. A first microstate free entropy χ(X 1 , ..., X n ) is defined as a normalized limit of the volume of sets of microstate i.e. matricial approximations (in moments) of the n-tuple of self-adjoints X i living in a (tracial) W * -probability space M. Starting from a definition of a free Fisher information [43], Voiculescu also defined a non-microstate free entropy χ * (X 1 , ..., X n ), known by the fundamental work [2] to be greater than the previous microstate entropy, and believed to be equal (at least modulo Connes' embedding conjecture). For more details, we refer the reader to the survey [45] for a list of properties as well as applications of free entropies in the theory of von Neumann algebras.Moreover in case of infinite entropy, two other invariants the microstate and nonmicrostate free entropy dimensions (respectively written δ 0 (X 1 , ..., X n ) and δ * (X 1 , ..., X n )) have been introduced to generalize results known for finite entropy. Surprisingly, Connes and Shlyakhtenko found in [10] a relation between those entropy dimensions and the first L 2 -Betti numbers they defined for finite von Neumann algebras. For instance, for (real and imaginary parts of ) generators of finitely generated groups, δ * has been proved in [27] to be equal to β-Betti numbers of groups). In [39], Dimitri Shlyakhtenko obtained lower bounds on microstate free entropy dimension (motivated by the goal of trying to prove equality with non-microstate free entropy dimension), in studying the following free stochastic differential equation :where ξ is the unique derivation densely defined (on non-commutative polynomials) such that ∂ j (X i ) = 1 i=j 1 ⊗ 1. Then the i-th conjugate variable is defined by 2000 Mathematics Subject Classification. 46L54, 60H15.

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Section: Introductionmentioning

confidence: 96%

A free stochastic partial differential equation

Dabrowski¹

2014

Ann. Inst. H. Poincaré Probab. Statist.

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“…Although, the axiomatic approach allows to write down the MME (13,14) in a mathematically consistent form, we generally need additional phenomenological insights which could lead to concrete form of the operators {V j }. The first problem is the relation between the dissipative part in (13,14) and the Hamiltonian of the open system.…”

Section: E Problems and Pitfallsmentioning

confidence: 99%

“…The first problem is the relation between the dissipative part in (13,14) and the Hamiltonian of the open system. Here the demanded structure of the stationary state and detailed balance condition are helpful [7].For example adding a nonlinear term to the Hamiltonian of the harmonic oscillator we have to modify the dissipative part too, in contrast to often used simplified models with linear dissipation.…”

Section: E Problems and Pitfallsmentioning

confidence: 99%

“…Obviously, as for any eigenfrequency ω the matrix [R kl (ω)] is positively defined we can always rewrite (65) in a diagonal form (13,14).…”

Section: Weak Coupling Limitmentioning

confidence: 99%

“…The MME (96) possesses the following well-known drawbacks: 1) the solution of (96) does not preserve positivity of the density matrix, 2) the Gibbs state (perhaps unnormalized) ∼ exp{−H S /T } is not its stationary state. The first drawback can be cured by adding the term −γ(8M T ) −1 [P, [P, ρ t ]] which allows to write the corrected MME in a standard form (13,14). However the dissipative part of this new generator corresponds to a damped harmonic oscillator one (27) (with γ ↑ = 0) and the Hamiltonian part gains the correction ∼ (XP + P X) with a rather unclear physical interpretation.…”

Section: Open Systems With Continuous Spectrummentioning

confidence: 99%

See 2 more Smart Citations

Invitation to Quantum Dynamical Semigroups

Alicki

2002

Dynamics of Dissipation

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The theory of quantum dynamical semigroups within the mathematically rigorous framework of completely positive dynamical maps is reviewed. First, the axiomatic approach which deals with phenomenological constructions and general mathematical structures is discussed. Then basic derivation schemes of the constructive approach including singular coupling, weak coupling and low density limits are presented in their higly simplified versions. Two-level system coupled to a heat bath, damped harmonic oscillator, models of decoherence, quantum Brownian particle and Bloch-Boltzmann equations are used as illustrations of the general theory. Physical and mathematical limitations of the quantum open system theory, the validity of Markovian approximation and alternative approaches are discussed also.

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Quantum Stochastic Differential Equations and Dilation of Completely Positive Semigroups

Fagnola

2006

Lecture Notes in Mathematics

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Sufficient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups

Cited by 115 publications

References 21 publications

A free stochastic partial differential equation

A free stochastic partial differential equation

Invitation to Quantum Dynamical Semigroups

Quantum Stochastic Differential Equations and Dilation of Completely Positive Semigroups

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