REVIEW
Bell inequalities: many questions, a few answers
Quantum Physics · Submitted 2 Feb 2007 · 68 references
Pith's verdict: A Festschrift review with one real technical addition: the AS_n family is worth checking, but its general-n local bound is asserted rather than proved.
T0 · AI review
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Referee report
The paper
The paper claims
Gisin lays out what is still unknown about Bell inequalities, and adds a new family of them for any even number of measurement settings.
The reading
What carries the argument
The machinery is the local polytope: for any fixed finite number of players, inputs and outcomes, the set of correlations admitting a shared-randomness local model is a convex polytope bounded by facets, and each facet is a tight Bell inequality. Correlation inequalities are the special case written only in terms of expectation values $E(x,y)$, which is what makes the AS$_n$ matrices compact. The figure of merit throughout is the visibility threshold $V_n$: the smallest admixture of white noise at which a maximally entangled state still violates the inequality, so that a lower $V_n$ means a more noise-tolerant test. Appendix B carries a second device, the Bechmann-Pasquinucci-Gisin inequality $S_{3 times 4}$, whose quantum maximum $4 sqrt 3$ exceeds the best real-amplitude value $2 + 2 sqrt 5$, making it a candidate separator between real and complex Hilbert spaces.
Extended reading notes
Core claim
The new result is the AS$_n$ family of bipartite correlation Bell inequalities, valid for any even number $n$ of inputs per side with binary outcomes. Each member is written as a matrix whose entries are coefficients on the expectation values $E(x,y) = p(a=b|x,y) - p(a=b|x,y)$; the displayed cases have local bounds $6$ for $n=4$, $12$ for $n=6$ and $20$ for $n=8$. The pattern is a first row of all $+1$, then successive rows shifted left with the falling entry re-entering with a changed coefficient, ending in a row $+1,-1,0,0,0,0$; $n=2$ reduces to CHSH. Gisin reports that these inequalities are tight and maximally violated by maximally entangled qubit states for visibilities above $V_2 = 1/2$, $V_4 = 0.7348$, $V_{10} = 0.7469$, $V_{50} = 0.7499$, apparently tending to $0.75$. That contrasts with the earlier Inn22 family, whose threshold appears to tend to $1$ as the number of inputs grows. All the settings can be chosen on a great circle of the Poincar sphere. A second, diagonal D-family found by numerical search for five and six inputs is listed in Appendix A.
Load-bearing premise
The step from the explicitly verified four-setting case to every even number of settings is asserted rather than proved: the paper calls the generalization straightforward and backs the local bounds and tightness with numerical search at a few values of $n$ rather than an analytic derivation.
Editorial extensions
If this is right
- Experimenters gain a tunable family that interpolates between the two-setting CHSH test and many-setting tests, so measurement settings can be traded against noise tolerance.
- Because the threshold visibility rises toward about $0.75$, the family certifies nonlocality for Werner states in a visibility range where CHSH, which stops at $1/sqrt 2$ approximately $0.7071$, cannot.
- The contrast with the Inn22 family, whose threshold appears to tend to $1$, shows that adding settings does not automatically buy noise tolerance; placing all settings on a great circle of the Poincar sphere is what makes the AS$_n$ family work.
- The $S_{3 times 4}$ inequality gives a concrete test that separates real-number quantum strategies from complex ones, provided the gap survives in Hilbert spaces larger than the one used to compute it.
Reading between the lines
- If the AS$_n$ thresholds really saturate near $0.75$, a gap remains between that value and the best known local model for Werner states, around $0.66$; no member of this family closes it, so a different construction would be needed.
- The local bounds $6, 12, 20$ grow quadratically in $n$; a proof of that scaling would likely double as the missing general-$n$ facet proof, and might also suggest the matching quantum strategy.
- The open-question list works as a checklist that later work can be scored against: POVM necessity, bound entanglement violating some Bell inequality, and real-versus-complex separations are all natural first targets for the new inequalities as test cases.
- The detection-loophole discussion suggests a follow-up calculation: compute the detection efficiency required by AS$_n$ for large $n$ and compare it with the $81.65$ percent achieved by the three-setting inequality of Massar and co-workers, to see whether the new family also helps in the laboratory.
Original abstract
What can be more fascinating than {\it experimental metaphysics}, to quote one of Abner Shimony's enlightening expressions? Bell inequalities are at the heart of the study of nonlocality. I present a list of open questions, organised in three categories: fundamental; linked to experiments; and exploring nonlocality as a resource. New families of inequalities for binary outcomes are presented.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
What can be more fascinating than {\it experimental metaphysics}, to quote one of Abner Shimony's enlightening expressions? Bell inequalities are at the heart of the study of nonlocality. I present a list of open questions, organised in three categories: fundamental; linked to experiments; and exploring nonlocality as a resource. New families of inequalities for binary outcomes are presented.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | What can be more fascinating than {\it experimental metaphysics}, to quote one of Abner Shimony's enlightening expressions? | background 1.00 | 0.03 | 0.02 | unclear 0.88 |
| s1 | Bell inequalities are at the heart of the study of nonlocality. | background 1.00 | 0.05 | 0.08 | asserted 1.00 |
| s2 | I present a list of open questions, organised in three categories: fundamental; linked to experiments; and exploring nonlocality as a resource. | contribution 0.96 | 0.89 | 0.19 | asserted 0.98 |
| s3 | New families of inequalities for binary outcomes are presented. | contribution 0.93 | 0.92 | 0.21 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:quant-ph/0702021 by the authors.
References
How later papers used it
63 later papers cite it
1 later paper by other authors is listed below: 1 used its method or tool.
Uses its method or tool 1 paper
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Based its construction on the Bell inequality introduced in this paper.
Optimal randomness certification from one entangled bit (2015) · 0.85 · paraphrased: the licence does not allow quoting
How the literature uses it
62 arXiv papers cite this in 105 sentences
Papers citing it · 82 sentences without a confident label
Builds on it 1 paper
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This generalises the ambiguity previously identified for the Elegant Bell inequality [ref] (recovered as the special case $n=3$), showing that it is not an isolated feature but a general algebraic consequence of Bell functionals whose optimal observables realise complex Clifford-algebra representations.
Scalable Self-Testing of Mutually Anticommuting Observables and Maximally Entangled Two-Qudits · Summary and Discussion · 0.98
Uses its method or tool 1 paper
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The construction is based on the elegant Bell inequality introduced in [ref] .
Optimal randomness certification from one entangled bit · Introduction · 0.85
Compares with it 4 papers
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Note, that the inequalities $X_2$ and $Y_2$ are in fact the CHSH inequality [ref] , whereas $X_3$ is just a member of the elegant Bell inequalities [ref] denoted by $S_3×4$ in Ref. [ref] .
Generalized Clauser-Horne-Shimony-Holt inequalities maximally violated by higher dimensional systems · New families of Bell inequalities · 0.88
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It is worth noting that the above inequality is similar to the general form of Gisin's "elegant Bell inequalities" discussed in [ref] .
Intraparticle entanglement-based Random Access Code protocols: Contextuality-enabled quantum advantage and implications · $n ↦1$ generalisation · 0.96
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The derived Bell inequality for a qubit SIC (which corresponds to a tetrahedron configuration on the Bloch sphere) can be compared to the so-called elegant Bell inequality [ref] whose maximal violation is also achieved using the tetrahedron configuration.
Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments · Bell inequalities for SICs · 0.97
Cites it as background 16 papers
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In addition to the implications for self-testing untrusted quantum apparatus, this also resolved an open question posed by Gisin [ref] related to the violation of Bell inequalities.
Generalized self-testing and the security of the 6-state protocol · Introduction · 0.81
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A number of authors have further investigated the nonlocal properties of Werner states [ref] .
Strong violations of Bell-type inequalities for Werner-like states · Introduction · 0.96
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The Abner Shimony (AS) inequalities are a family of tight two-qubit $n$-setting ($n$ is even) Bell inequalities [ref] [ref] .
General Hardy-Type Paradox Based on Bell inequality and its Experimental Test · General Hardy-type paradox based on the AS inequality · 0.97
Discussion
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