Derivation of Hartreeʼs theory for generic mean-field Bose systems

Lewin, Mathieu; Nam, Phan Thành; Rougerie, Nicolas

doi:10.1016/j.aim.2013.12.010

Cited by 127 publications

(49 citation statements)

References 63 publications

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“…Since {γ (k) ∞,t } is obtained as a weak * limit of a BBGKY sequence, it verifes the assumption of the weak quantum de Finetti theorem in [32,9,29,18,40]. That…”

Section: Energy Estimates and A-priori Bounds On γ Nt = {γsupporting

confidence: 50%

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Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

Yuan¹

2015

Communications on Pure &Amp; Applied Analysis

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In this paper, we investigate the dynamics of a boson gas with three-body interactions in T 2 . We prove that when the particle number N tends to infinity, the BBGKY hierarchy of k-particle marginals converges to a infinite Gross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions, and for the asymptotically factorized N -body initial datum, we show that this N → ∞ limit corresponds to the quintic nonlinear Schrödinger equation. Thus, the Bose-Einstein condensation is preserved in time.2000 Mathematics Subject Classification. Primary: 35L15, 35L45; Secondary: 35Q40.

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“…Since {γ (k) ∞,t } is obtained as a weak * limit of a BBGKY sequence, it verifes the assumption of the weak quantum de Finetti theorem in [32,9,29,18,40]. That…”

Section: Energy Estimates and A-priori Bounds On γ Nt = {γsupporting

confidence: 50%

“…[37,39,33,34,35,1,2,4,5,6,7,8,20,22,23,24,21,10,18,26,28,36,31,30,25,29,32,38,40], and references therein. A fundamental problem is to prove that Bose-Einstein condensation occurs for such systems.…”

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confidence: 99%

Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

Yuan¹

2015

Communications on Pure &Amp; Applied Analysis

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“…These quantum de Finetti theorems are appealing not only due to their own elegance on the characterization of symmetric states, but also because of the successful applications in many-body physics [5,11,12], quantum information [9,13,14], and computational complexity theory [10,15,16].…”

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confidence: 99%

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

Smith²

2015

Phys. Rev. Lett.

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We prove a version of the quantum de Finetti theorem: permutation-invariant quantum states are well approximated as a probabilistic mixture of multifold product states. The approximation is measured by distinguishability under measurements that are implementable by fully-one-way local operations and classical communication (LOCC). Our result strengthens Brandão and Harrow's de Finetti theorem where a kind of partially-one-way LOCC measurements was used for measuring the approximation, with essentially the same error bound. As main applications, we show (i) a quasipolynomial-time algorithm which detects multipartite entanglement with an amount larger than an arbitrarily small constant (measured with a variant of the relative entropy of entanglement), and (ii) a proof that in quantum Merlin-Arthur proof systems, polynomially many provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations.

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“…For the symmetric case, we denote the corresponding joint product numerical range by Π + (H 1 , H 2 , H 3 ), and the joint separable numerical range by Θ + (H 1 , H 2 , H 3 ). Physically, we are dealing with a many-body bosonic system with symmetric wavefunctions in the N → ∞ limit, where the reduced density matrices of the wave function of the system is also known to be separable due to the quantum de Finetti's theorem [21]. We consider Π + (H 1 , H 2 , H 3 ) that is given by the set of points (x, y, z) ∈ R 3 , where…”

Section: The Symmetric Case and Bosonic Systemsmentioning

confidence: 99%

“…We will develop a method that systematically leads to many other possibilities of ruled surface for the threedimensional projections of 2-RDMs. We start from a fact that although the geometry of 2-RDMs are in general hard to characterize, there is one situation it is provably easy: that is, for an infinite spatial dimensional system, the 2-RDMs are known to be separable, due to the celebrated quantum de Finetti's theorem [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Joint product numerical range and geometry of reduced density matrices

Chen

Guo

et al. 2016

Sci. China Phys. Mech. Astron.

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The reduced density matrices of a many-body quantum system form a convex set, whose threedimensional projection Θ is convex in R 3 . The boundary ∂Θ of Θ may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces: symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti's theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range Π of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that ruled surface emerge naturally when taking a convex hull of Π. We show that, a ruled surface on ∂Θ sitting in Π has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of Θ, with two boundary pieces of symmetry breaking origin separated by two gapless lines.

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Derivation of Hartreeʼs theory for generic mean-field Bose systems

Cited by 127 publications

References 63 publications

Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

Joint product numerical range and geometry of reduced density matrices

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