Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Europhysics Letters (EPL)

De Finetti's Theorem and Bell-Type Correlation Inequalities

Alexander Bach

Published Oct 7, 1991• 2 citations
Publisher

Abstract

The quantum version of de Finetti's theorem implies the existence of a large class of correlation inequalities for indistinguishable particles, e.g. Bell's inequalities. Systems which violate these inequalities are in a state that is compatible with a finite number of particles only.
DOI
10.1209/0295-5075/16/6/001

Related papers

Bell's inequality violations: relation with de Finetti's coherence principle and inferential analysis of experimental data

Franco Fagnola, Matteo Gregoratti

It is often believed that de Finetti's coherence principle naturally leads, in the finite case, to the Kolmogorov's probability theory of random phenomena, which then implies Bell's inequality.Thus, not only a violation of Bell's inequality looks paradoxical in the Kolmogorovian framework, but it should violate also de Finetti's coherence principle.Firstly, we show that this is not the case: the typical theoretical violations of Bell's inequality in quantum physics are in agreement with de Finetti's coherence principle.Secondly, we look for statistical evidence of such violations: we consider the experimental data of measurements of polarization of photons, performed to verify empirically violations of Bell's inequality, and, on the basis of the estimated violation, we test the null hypothesis of Kolmogorovianity for the observed phenomenon.By standard inferential techniques we compute the pvalue for the test and get a clear strong conclusion against the Kolmogorovian hypothesis.

Communications on Stochastic Analysis•Mar 1, 2012•1 citation

de Finetti reductions for correlations

Rotem Arnon-Friedman, Renato Renner

When analysing quantum information processing protocols, one has to deal with large entangled systems, each consisting of many subsystems. To make this analysis feasible, it is often necessary to identify some additional structures. de Finetti theorems provide such a structure for the case where certain symmetries hold. More precisely, they relate states that are invariant under permutations of subsystems to states in which the subsystems are independent of each other. This relation plays an important role in various areas, e.g., in quantum cryptography or state tomography, where permutation invariant systems are ubiquitous. The known de Finetti theorems usually refer to the internal quantum state of a system and depend on its dimension. Here, we prove a different de Finetti theorem where systems are modelled in terms of their statistics under measurements. This is necessary for a large class of applications widely considered today, such as device independent protocols, where the underlying systems and the dimensions are unknown and the entire analysis is based on the observed correlations.

arXiv:1308.0312•Journal of Mathematical Physics•May 1, 2015•20 citations

Remarks on the Quantum de Finetti Theorem for Bosonic Systems

Mathieu Lewin B, Phan Thành Nam, Nicolas Rougerie

The quantum de Finetti theorem asserts that the |$k$|-body density matrices of an |$N$|-body bosonic state approach a convex combination of Hartree states (pure tensor powers) when |$N$| is large and |$k$| fixed. In this note, we review a construction due to Christandl et al. [8] valid for finite-dimensional Hilbert spaces, which gives a quantitative version of the theorem. We first propose a variant of their proof that leads to a slightly improved estimate. Next we provide an alternative proof of an explicit formula due to Chiribella [7], which gives the density matrices of the constructed state as a function of those of the original state.

arXiv:1310.2200•Applied Mathematics Research eXpress•Oct 9, 2014•34 citations

Bell-Type Inequalities for Partial Separability in N -Particle Systems and Quantum Mechanical Violations

Michael Seevinck, George Svetlichny

We derive N-particle Bell-type inequalities under the assumption of partial separability, i.e., that the N-particle system is composed of subsystems which may be correlated in any way (e.g., entangled) but which are uncorrelated with respect to each other. These inequalities provide, upon violation, experi-mentally accessible sufficient conditions for full N-particle entanglement, i.e., for N-particle entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. The inequalities are shown to be maximally violated by the N-particle Greenberger-Horne-Zeilinger states.

arXiv:quant-ph/0201046•Physical Review Letters•Jul 18, 2002•216 citations

Quantum de Finetti theorem in phase-space representation

Anthony Leverrier, Nicolas J. Cerf

The quantum versions of de Finetti's theorem derived so far express the convergence of $n$-partite symmetric states, i.e., states that are invariant under permutations of their $n$ parties, toward probabilistic mixtures of independent and identically distributed (IID) states of the form ${\ensuremath{\sigma}}^{\ensuremath{\bigotimes}n}$. Unfortunately, these theorems only hold in finite-dimensional Hilbert spaces, and their direct generalization to infinite-dimensional Hilbert spaces is known to fail. Here, we address this problem by considering invariance under orthogonal transformations in phase space instead of permutations in state space, which leads to a quantum de Finetti theorem particularly relevant to continuous-variable systems. Specifically, an $n$-mode bosonic state that is invariant with respect to this continuous symmetry in phase space is proven to converge toward a probabilistic mixture of IID Gaussian states (actually, $n$ identical thermal states).

arXiv:0904.4862•Physical Review A•Jul 28, 2009•11 citations

On Bell's Theorem

Jorma Jormakka

Bell's Theorem implies that quantum correlation of entangled particles as calculated in quantum mechanics violates elementary probabilistic inequalities. It is shown that the reason is a problem in scaling of detector directions.

•Oct 1, 2018•0 citations
It is often believed that de Finetti's coherence principle naturally leads, in the finite case, to the Kolmogorov's probability theory of random phenomena, which then implies Bell's inequality.Thus, not only a violation of Bell's inequality looks paradoxical in the Kolmogorovian framework, but it should violate also de Finetti's coherence principle.Firstly, we show that this is not the case: the typical theoretical violations of Bell's inequality in quantum physics are in agreement with de Finetti's coherence principle.Secondly, we look for statistical evidence of such violations: we consider the experimental data of measurements of polarization of photons, performed to verify empirically violations of Bell's inequality, and, on the basis of the estimated violation, we test the null hypothesis of Kolmogorovianity for the observed phenomenon.By standard inferential techniques we compute the pvalue for the test and get a clear strong conclusion against the Kolmogorovian hypothesis.
When analysing quantum information processing protocols, one has to deal with large entangled systems, each consisting of many subsystems. To make this analysis feasible, it is often necessary to identify some additional structures. de Finetti theorems provide such a structure for the case where certain symmetries hold. More precisely, they relate states that are invariant under permutations of subsystems to states in which the subsystems are independent of each other. This relation plays an important role in various areas, e.g., in quantum cryptography or state tomography, where permutation invariant systems are ubiquitous. The known de Finetti theorems usually refer to the internal quantum state of a system and depend on its dimension. Here, we prove a different de Finetti theorem where systems are modelled in terms of their statistics under measurements. This is necessary for a large class of applications widely considered today, such as device independent protocols, where the underlying systems and the dimensions are unknown and the entire analysis is based on the observed correlations.
The quantum de Finetti theorem asserts that the |kkk|-body density matrices of an |NNN|-body bosonic state approach a convex combination of Hartree states (pure tensor powers) when |NNN| is large and |kkk| fixed. In this note, we review a construction due to Christandl et al. [8] valid for finite-dimensional Hilbert spaces, which gives a quantitative version of the theorem. We first propose a variant of their proof that leads to a slightly improved estimate. Next we provide an alternative proof of an explicit formula due to Chiribella [7], which gives the density matrices of the constructed state as a function of those of the original state.
We derive N-particle Bell-type inequalities under the assumption of partial separability, i.e., that the N-particle system is composed of subsystems which may be correlated in any way (e.g., entangled) but which are uncorrelated with respect to each other. These inequalities provide, upon violation, experi-mentally accessible sufficient conditions for full N-particle entanglement, i.e., for N-particle entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. The inequalities are shown to be maximally violated by the N-particle Greenberger-Horne-Zeilinger states.
The quantum versions of de Finetti's theorem derived so far express the convergence of nnn-partite symmetric states, i.e., states that are invariant under permutations of their nnn parties, toward probabilistic mixtures of independent and identically distributed (IID) states of the form ${\ensuremath{\sigma}}^{\ensuremath{\bigotimes}n}$. Unfortunately, these theorems only hold in finite-dimensional Hilbert spaces, and their direct generalization to infinite-dimensional Hilbert spaces is known to fail. Here, we address this problem by considering invariance under orthogonal transformations in phase space instead of permutations in state space, which leads to a quantum de Finetti theorem particularly relevant to continuous-variable systems. Specifically, an nnn-mode bosonic state that is invariant with respect to this continuous symmetry in phase space is proven to converge toward a probabilistic mixture of IID Gaussian states (actually, nnn identical thermal states).