A de Finetti-type theorem withm-dependent states

Petz, Dénes

doi:10.1007/bf01377629

Cited by 14 publications

(12 citation statements)

References 14 publications

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“…There have been many applications of the de Finetti theorem to topics including foundational issues [7,26], mathematical physics [17,27] and quantum information theory [10,18,28,29,30,31]; there have also been various generalisations [3,4,5,6,7,9,10,15,16,17]. We…”

Section: Discussionmentioning

confidence: 99%

“…, x k ) = P X (x 1 ) · · · P X (x k )dµ(P X ), (1) where µ is a measure on the set of probability distributions, P X , of one variable. In the quantum analogue [3,4,5,6,7,8] a state ρ k on H ⊗k is said to be infinitely exchangeable if it is symmetric (or permutationinvariant), i.e. πρ k π † = ρ k for all π ∈ S k and, for all n > k, there is a symmetric state ρ n on H ⊗n with ρ k = tr n−k ρ n .…”

Section: Introductionmentioning

confidence: 99%

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One-and-a-Half Quantum de Finetti Theorems

Christandl

König

Mitchison

et al. 2007

Commun. Math. Phys.

174

294

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When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by 2 kd 2 n , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation Uµ+ν ⊂ Uµ ⊗ Uν of the unitary group U(d), for highest weights µ, ν and µ + ν. Let ξµ be the state obtained by tracing out Uν. Then ξµ is close to a convex combination of the coherent states Uµ(g)|vµ , where g ∈ U(d) and |vµ is the highest weight vector in Uµ.For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our "half" theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

One-and-a-Half Quantum de Finetti Theorems

Christandl

König

Mitchison

et al. 2007

Commun. Math. Phys.

174

294

View full text Add to dashboard Cite

show abstract

“…We will be interested in the set of states that minimize the energy density associated to H in the limit N → ∞. Such a set can be easily characterized [24] by making use of the quantum de Finetti theorem [1][2][3][4][5][6][7][8][9]. This theorem states that the two-spin reduced states of any state that is symmetric under permutations can be written as a convex combinations of product states, σ = µ ⊗ µ, in the limit N → ∞.…”

Section: Symmetric Hamiltonians In Spin Latticesmentioning

confidence: 99%

“…The success of this approach can be understood from different perspectives: (i) For spin systems with few-body interactions that are invariant under permutations, the exact ground state is a product state in the thermodynamic limit, where the number of spins N → ∞. This can be viewed as a direct consequence of the quantum de Finetti theorem [1][2][3][4][5][6][7][8][9]: It states that any density operator of L spins, σ, with a symmetric extension in the thermodynamic limit, is separable and can be written as a convex combination of product states. (ii) If we consider regular spin lattices in d → ∞ dimensions with nearest-neighbor interactions and translation and rotation symmetry, monogamy of entanglement [10,11] implies that the ground state wave functions have to be of product type.…”

Section: Introductionmentioning

confidence: 99%

Ground states of fermionic lattice Hamiltonians with permutation symmetry

2013

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We study the ground states of lattice Hamiltonians that are invariant under permutations, in the limit where the number of lattice sites, N → ∞. For spin systems, these are product states, a fact that follows directly from the quantum de Finetti theorem. For fermionic systems, however, the problem is very different, since mode operators acting on different sites do not commute, but anti-commute. We construct a family of fermionic states, F, from which such ground states can be easily computed. They are characterized by few parameters whose number only depends on M , the number of modes per lattice site. We also give an explicit construction for M = 1, 2. In the first case, F is contained in the set of Gaussian states, whereas in the second it is not. Inspired by that constructions, we build a set of fermionic variational wave functions, and apply it to the Fermi-Hubbard model in two spatial dimensions, obtaining results that go beyond the generalized Hartree-Fock theory.

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“…There are various noncommutative versions of this theorem [1,10,11,16,24], all of which involve tensor product constructions or other commutativity conditions. Indeed there is no hope to obtain a general De Finetti's theorem without imposing additional conditions.…”

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confidence: 99%

Cumulants in noncommutative probability theory IV. Noncrossing cumulants: De Finetti's theorem and Lp-inequalities

Lehner

2006

Journal of Functional Analysis

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De Finetti's theorem states that any exchangeable sequence of classical random variables is conditionally i.i.d. with respect to some σ -algebra. In this paper we prove a "free" noncommutative analog of this theorem, namely we show that any noncrossing exchangeability system with a faithful state which satisfies a so called weak singleton condition can be embedded into an free product with amalgamation over a certain subalgebra such that the interchangeable algebras remain interchangeable with respect to the operator-valued expectation. Vanishing of crossing cumulants can be verified by checking a certain weak freeness condition and the weak singleton condition is satisfied e.g. when the state is tracial. The proof follows the classical proof of De Finetti's theorem, the main technical tool being a noncommutative L p -inequality for i.i.d. sums of centered noncommutative random variables in noncrossing exchangeability systems.

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A de Finetti-type theorem withm-dependent states

Cited by 14 publications

References 14 publications

One-and-a-Half Quantum de Finetti Theorems

One-and-a-Half Quantum de Finetti Theorems

Ground states of fermionic lattice Hamiltonians with permutation symmetry

Cumulants in noncommutative probability theory IV. Noncrossing cumulants: De Finetti's theorem and Lp-inequalities

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