Bell inequalities for three systems and arbitrarily many measurement outcomes

Grandjean, Basile; Liang, Yeong-Cherng; Bancal, Jean-Daniel; Brunner, Nicolás; Gisin, Nicolas

doi:10.1103/physreva.85.052113

Cited by 21 publications

(26 citation statements)

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“…The families of Bell-like inequalities presented in [21,74] are some possible starting points for such an investigation and the numerical techniques that we detailed in Appendix G will be useful for this purpose. Note also that for any given positive integer k, Theorem 2 of [51] allows us to extend any given witness for n ≥ k parties to one for arbitrarily many parties while preserving the number of expectation values that need to be measured experimentally.…”

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confidence: 99%

Family of Bell-like Inequalities as Device-Independent Witnesses for Entanglement Depth

et al. 2015

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We present a simple family of Bell inequalities applicable to a scenario involving arbitrarily many parties, each of which performs two binary-outcome measurements. We show that these inequalities are members of the complete set of full-correlation Bell inequalities discovered by Werner-WolfZukowski-Brukner. For scenarios involving a small number of parties, we further verify that these inequalities are facet-defining for the convex set of Bell-local correlations. Moreover, we show that the amount of quantum violation of these inequalities naturally manifests the extent to which the underlying system is genuinely many-body entangled. In other words, our Bell inequalities, when supplemented with the appropriate quantum bounds, naturally serve as device-independent witnesses for entanglement depth, allowing one to certify genuine k-partite entanglement in an arbitrary n ≥ k-partite scenario without relying on any assumption about the measurements being performed, nor the dimension of the underlying physical system. A brief comparison is made between our witnesses and those based on some other Bell inequalities, as well as the quantum Fisher information. A family of witnesses for genuine k-partite nonlocality applicable to an arbitrary n ≥ k-partite scenario based on our Bell inequalities is also presented. One of the most important no-go theorems in physics concerns the impossibility to reproduce all quantum mechanical predictions using any locally-causal theory [1] -a fact commonly referred to as Bell's theorem [2]. An important observation leading to this celebrated result is that measurement statistics allowed by such theories must satisfy constraints in the form of an inequality, a Bell inequality. Since these inequalities only involve experimentally accessible quantities, their violation -a manifestation of Bell-nonlocality [3] -can be, and has been (modulo some arguably implausible loopholes [4]) empirically demonstrated (see, e.g., [3][4][5] and references therein).Clearly, Bell inequalities played an instrumental role in the aforementioned discovery. Remarkably, they also find applications in numerous quantum information and communication tasks, e.g., in quantum key distribution involving untrusted devices [6][7][8], in the reduction of communication complexity [9], in the expansion of trusted random numbers [10,11], in certifying the Hilbert space dimension of physical systems [12,13], in self-testing [14-18] of quantum devices, in witnessing [19][20][21] and quantifying [22-25] (multipartite) quantum entanglement using untrusted devices etc. For a recent review on these and other applications, see [3].Identifying interesting or useful Bell inequalities is nonetheless by no means obvious. For instance, the approach of solving for the complete set of optimal, i.e., facet-defining Bell inequalities for a given experimental scenario -though potentially useful for the identification of non-Bell-local (hereafter nonlocal) correlationstypically produces a large number of inequalities with no * yliang@phys.ethz....

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Family of Bell-like Inequalities as Device-Independent Witnesses for Entanglement Depth

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“…In contrast, some of these Bell inequalities that are reducible to Bell inequalities involving fewer number of outcomes require entangled two-qutrit states to achieve maximal quantum violation. These results show, once again (see e.g., [33,39,41]), that in analyzing the quantum violation of given Bell inequalities, the widely employed wisdom of setting the local Hilbert space dimension to be the same as the number of measurement outcomes is unfounded.…”

Section: Discussionmentioning

confidence: 90%

“…Interestingly, some of these newly obtained Bell inequalities-despite being ternary-outcome and irreducible to one having fewer measurement outcomes-can already be violated maximally via local measurements on entangled twoqubit states. Our work thus complements that of [33,39,41], showing that in determining the quantum state that maximally violates a given Bell inequality, optimal choice of the local Hilbert space dimension is not necessarily correlated with the (maximal) number of measurement outcomes involved.…”

Section: Introductionmentioning

confidence: 90%

“…To date, however, the bulk of such studies have focussed on the simplest Clauser-Horne-Shimony-Holt (CHSH) [26] Bell scenario, namely, one involving only two parties, each performing two binary-outcome measurements. While more complicated Bell scenarios, such as those involving more parties [25,27,28], or more measurement settings [29][30][31][32][33] or more measurement outcomes [34][35][36] have also been individually considered, scenarios involving a combination of these have so far received relatively little attention (see, however, [21,[36][37][38][39]). …”

Section: Introductionmentioning

confidence: 99%

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Bipartite Bell inequalities with three ternary-outcome measurements—from theory to experiments

et al. 2016

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“…CHSH inequalities [4][5][6] which are not fulfilled by all states of quantum mechanics are examples of Bell inequalities. Several other families of Bell inequalities have been derived in different cases [7][8][9][10][11][12][13][14][15][16][17].…”

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confidence: 99%

Bell inequalities for arbitrary situations

Heshmatpour

2015

Physics Letters A

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Bell inequalities for three systems and arbitrarily many measurement outcomes

Cited by 21 publications

References 54 publications

Family of Bell-like Inequalities as Device-Independent Witnesses for Entanglement Depth

Family of Bell-like Inequalities as Device-Independent Witnesses for Entanglement Depth

Bipartite Bell inequalities with three ternary-outcome measurements—from theory to experiments

Bell inequalities for arbitrary situations

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