Non-symmetric Dirichlet Forms on Semifinite von Neumann Algebras

Guido, Daniele; Isola, Tommaso; Scarlatti, Sergio

doi:10.1006/jfan.1996.0003

Cited by 22 publications

(11 citation statements)

References 27 publications

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“…We shall now built a non-commutative functional calculus associated to the derivations ∂ j . This functional calculus can either be seen as a generalization of the one developed in [13] (see also [20]), in the context of non-commutative Dirichlet form on C * -algebras, or as a generalization of the chain rule formula in [12]. It is based on the following representation of the space of continuous functions on some interval I on B(H).…”

Section: P -Regularity Of the Dirichlet Form Under Strong Detailed Ba...mentioning

confidence: 99%

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Estimating the decoherence time using non-commutative Functional Inequalities

Bardet

2017

Preprint

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We generalize the notions of the non-commutative Poincaré and modified log-Sobolev inequalities for primitive quantum Markov semigroups (QMS) to not necessarily primitive ones. These two inequalities provide estimates on the decoherence time of the evolution. More precisely, we focus on an algebraic definition of environment-induced decoherence in open quantum systems which happens to be generic on finite dimensional systems and describes the asymptotic behavior of any QMS. An essential tool in our analysis is the explicit structure of the decoherence-free algebra generated by the QMS, a central object in the study of passive quantum error correction schemes. The Poincaré constant corresponds to the spectral gap of the QMS, which implies its positivity, while we prove that the modified log-Sobolev constant is positive under the L1-regularity of the Dirichlet form, a condition that also appears in the primitive case. We furthermore prove that strong Lp-regularity holds for quantum Markov semigroups that satisfy a strong form of detailed balance condition for p ≥ 1. The latter condition includes all known cases where this strong regularity was proved. Finally and to emphasize the mathematical interest of this study compared to the classical case, we focus on two truly quantum scenarios, one exhibiting quantum coherence, and the other, quantum correlations.

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Section: P -Regularity Of the Dirichlet Form Under Strong Detailed Ba...mentioning

confidence: 99%

“…• When X = Y , Equation (4.7) is just the well-known chain rule formula for derivations as studied in [13,20]. We should simply write π X instead of π X,Y in this case.…”

Section: P -Regularity Of the Dirichlet Form Under Strong Detailed Ba...mentioning

confidence: 99%

Estimating the decoherence time using non-commutative Functional Inequalities

Bardet

2017

Preprint

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“…ρ 1/2 θG * θ = Gρ 1/2 , 2. the closed linear span of ρ 1/2 θL * ℓ θ | ℓ ≥ 1 and L ℓ ρ 1/2 | ℓ ≥ 1 in the Hilbert space of Hilbert-Schmidt operators on h coincide, 3. the self-adjoint trace class operators R, C defined by (17) and (22) commute and C −1 R is unitary and self-adjoint.…”

Section: Proposition 19 a Qms T Satisfies The Sqdb-θ If And Only If Tmentioning

confidence: 99%

“…Their non-commutative counterpart has also been deeply investigated (Albeverio and Goswami [1], Cipriani [6], Davies and Lindsay [8], Goldstein and Lindsay [15], Guido, Isola and Scarlatti [17], Park [23], Sauvageot [26] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Generators of KMS Symmetric Markov Semigroups on $${\mathcal{B}({\rm h})}$$ Symmetry and Quantum Detailed Balance

Fagnola

Umanità

2010

Commun. Math. Phys.

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We find the structure of generators of norm continuous quantum Markov semigroups on B(h) that are symmetric with respect to the scalar product tr(ρ 1/2 x * ρ 1/2 y) induced by a faithful normal invariant state invariant state ρ and satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the socalled standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.

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“…(5) The important point to note here is the role of the Hilbert space L 2 (M). Namely, a large class of dynamical maps is defined in terms of Dirichlet forms on such Hilbert space, see [27], and [11] for more details and an comprehensive bibliography. Thus, this case will be treated in a separate section.…”

Section: Quantum Maps On the Set Of Regular Observablesmentioning

confidence: 99%

QUANTUM L_p AND ORLICZ SPACES

Labuschagne

Majewski

2008

Quantum Probability and Related Topics

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Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [46] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces L cosh −1 , L log(L + 1) . The present paper therefore in some sense "completes" the picture by showing that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition, canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation technique, specifically suited to the above context, for extending the action of such maps to the appropriate intermediate spaces of the pair L ∞ , L 1 . Moreover, it is shown that quantum dynamics in the form of Markov semigroups described by some Dirichlet forms naturally extends to the context proposed in [46].

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Non-symmetric Dirichlet Forms on Semifinite von Neumann Algebras

Cited by 22 publications

References 27 publications

Estimating the decoherence time using non-commutative Functional Inequalities

Estimating the decoherence time using non-commutative Functional Inequalities

Generators of KMS Symmetric Markov Semigroups on $${\mathcal{B}({\rm h})}$$ Symmetry and Quantum Detailed Balance

QUANTUM L_p AND ORLICZ SPACES

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Non-symmetric Dirichlet Forms on Semifinite von Neumann Algebras

Cited by 22 publications

References 27 publications

Estimating the decoherence time using non-commutative Functional Inequalities

Estimating the decoherence time using non-commutative Functional Inequalities

Generators of KMS Symmetric Markov Semigroups on $${\mathcal{B}({\rm h})}$$ Symmetry and Quantum Detailed Balance

QUANTUM Lp AND ORLICZ SPACES

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QUANTUM L_p AND ORLICZ SPACES