A Family of Monotone Quantum Relative Entropies

Lewin, Mathieu; Sabin, Julien

doi:10.1007/s11005-014-0689-y

Cited by 15 publications

(49 citation statements)

References 18 publications

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“…Then, in Sect. 6 we define the relative entropy of two density matrices following [33] and prove Lieb-Thirring inequalities in the spirit of [17,18]. We are able to deal with much more than the three examples (7)- (9).…”

Section: X)ρ(t Y) DX Dymentioning

confidence: 99%

“…The relative entropy H is properly defined and studied at length in [33] where we even considered a general concave function S, leading to many other stationary states γ f for the Hartree equation. We do not discuss this here for shortness.…”

Section: Bose and Fermi Gases At Positive Temperaturementioning

confidence: 99%

“…We use a general concept of relative entropy H(γ , γ f ) of two one-particle density matrices γ and γ f which has been recently introduced in [33]. The purpose of this section is to prove some useful lower bounds on this relative entropy.…”

Section: Inequalities On the Relative Entropymentioning

confidence: 99%

“…In [33], the relative entropy (75) was studied for S a general concave function, and several properties were proved. First by Klein's lemma (see [36,Prop.…”

Section: Inequalities On the Relative Entropymentioning

confidence: 99%

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The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory

Lewin

Sabin

2014

Commun. Math. Phys.

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123

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We show local and global well-posedness results for the Hartree equationwhere γ is a bounded self-adjoint operator on L 2 (R d ), ρ γ (x) = γ (x, x) and w is a smooth short-range interaction potential. The initial datum γ (0) is assumed to be a perturbation of a translation-invariant state γ f = f (− ) which describes a quantum system with an infinite number of particles, such as the Fermi sea at zero temperature, or the Fermi-Dirac and Bose-Einstein gases at positive temperature. Global well-posedness follows from the conservation of the relative (free) energy of the state γ (t), counted relatively to the stationary state γ f . We indeed use a general notion of relative entropy, which allows us to treat a wide class of stationary states f (− ). Our results are based on a Lieb-Thirring inequality at positive density and on a recent Strichartz inequality for orthonormal functions, which are both due to Frank, Lieb, Seiringer and the first author of this article.

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Section: X)ρ(t Y) DX Dymentioning

confidence: 99%

Section: Bose and Fermi Gases At Positive Temperaturementioning

confidence: 99%

Section: Inequalities On the Relative Entropymentioning

confidence: 99%

“…In [33], the relative entropy (75) was studied for S a general concave function, and several properties were proved. First by Klein's lemma (see [36,Prop.…”

Section: Inequalities On the Relative Entropymentioning

confidence: 99%

See 2 more Smart Citations

The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory

Lewin

Sabin

2014

Commun. Math. Phys.

Self Cite

123

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show abstract

“…For example, Tropp used the joint convexity of the quantum relative entropy -which is a special Bregman divergence -to give a succinct proof of a famous concavity theorem of Lieb [TR12]. In [LS14], Lewin and Sabin characterized a certain monotonicity property of the Bregman divergence by the operator monotonicity of the derivative of the corresponding scalar function. In [BB01], Bauschke and Borwein gave a necessary and sufficient condition for the joint convexity of the Bregman divergences on R d .…”

Section: Introductionmentioning

confidence: 99%

On the Joint Convexity of the Bregman Divergence of Matrices

Pitrik

Virosztek

2015

Lett Math Phys

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We characterize the functions for which the corresponding Bregman divergence is jointly convex on matrices. As an application of this characterization, we derive a sharp inequality for the quantum Tsallis entropy of a tripartite state, which can be considered as a generalization of the strong subadditivity of the von Neumann entropy. (In general, the strong subadditivity of the Tsallis entropy fails for quantum states, but it holds for classical states.) Furthermore, we show that the joint convexity of the Bregman divergence does not imply the monotonicity under stochastic maps, but every monotone Bregman divergence is jointly convex.

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Bose–Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature

Deuchert

Seiringer

Yngvason

2018

Commun. Math. Phys.

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We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross-Pitaevskii (GP) limit where the trap frequency ω, the temperature T and the particle number N are related by N ∼ (T /ω) 3 → ∞ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap.We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose-Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.2.4. The condensate energy and the interaction between the condensate and the thermal cloud .

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A Family of Monotone Quantum Relative Entropies

Cited by 15 publications

References 18 publications

The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory

The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory

On the Joint Convexity of the Bregman Divergence of Matrices

Bose–Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature

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