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Polynomial Bell inequalities

Rafael Chaves

arXiv:1506.04325v2quant-ph

TL;DR

General Bayesian-network Bell scenarios are characterized by polynomial inequalities, but methods to derive them have been limited. The paper adapts Fourier–Motzkin elimination to derive such inequalities, illustrates the method in several networks, and extends the construction to a notion of nonsignalling in generalized networks.

  • Problem

    Generalized local hidden-variable models have non-convex correlation sets characterized by polynomial Bell inequalities, while general-purpose quantifier-elimination methods are computationally intractable even for simple Bell scenarios.

    Source: §II · p. 3

  • Method

    The authors adapt Fourier–Motzkin elimination to combine linear probability constraints with nonlinear independence constraints and eliminate non-observable quantities.

    Source: §II · p. 3, Appendix A · p. 6

  • Results

    The method derives polynomial inequalities in several GLHV scenarios, including a three-setting bilocal example that detects non-bilocality for v > 4/9, compared with nonlocality only for v > 5/9 without the independence constraint.

    Source: §II · p. 4

  • Takeaways & Limitations

    The construction also gives a notion of nonsignalling correlations in generalized Bayesian networks by replacing classical hidden variables with general nonsignalling distributions.

    Source: §III · p. 4, §IV · p. 5

    Although the procedure can provide a full characterization, computational tractability may limit practical derivations to partial characterizations, such as full correlators only.

    Source: §II · p. 3

Abstract

from arXiv · show

It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the tool of Bayesian networks, a graphical model used to reason about probabilistic causation. Indeed, Bell's theorem concerns a particular kind of a Bayesian network and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalized Bell scenarios involving more complex Bayesian networks. The problem, however, relies on the fact that such generalized scenarios are characterized by polynomial Bell inequalities and no current method is available to derive them beyond very simple cases. In this work, we make a significant step in that direction, providing a general and practical method for the derivation of polynomial Bell inequalities in a wide class of scenarios, applying it to a few cases of interest. We also show how our construction naturally gives rise to a notion of non-signalling in generalized networks.

I. BELL INEQUALITIES, BAYESIAN NETWORKS AND MARGINAL PROBLEMS

Bayesian networks encode causal relations as conditional independencies, while Bell inequalities can be derived as constraints on observable marginals of a valid joint distribution. Fourier–Motzkin elimination removes non-observable quantities, recovering familiar Bell constraints such as CHSH.

I. BELL INEQUALITIES, BAYESIAN NETWORKS AND MARGINAL PROBLEMS

  • A Bayesian-network DAG represents variables and causal arrows, and its conditional independencies follow from d-separation.For an LHV model, the hidden variable is independent of the settings, and each party’s outcome depends only on its own setting and the hidden variable.Source: §I · p. 2
  • Bell inequalities can be treated as a marginal problem: whether specified observable distributions are marginals of a valid joint distribution over all possible outcomes.The joint distribution is constrained by positivity and normalization; eliminating its non-observable terms yields constraints on the observable distributions.Source: §I · p. 2
  • For dichotomic outcomes, the joint-probability description can be converted to a correlation vector whose components are expectation values.The correlation vector obeys linear inequalities obtained by transforming the probability constraints.Source: §I · p. 2
  • In the CHSH example, Fourier–Motzkin elimination removes non-observable correlators and recovers the CHSH inequality.The sum of two inequalities eliminates the remaining non-observable term.Source: §I · p. 2

II. POLYNOMIAL BELL INEQUALITIES

The paper adapts Fourier–Motzkin elimination to derive polynomial Bell inequalities for Bayesian networks whose joint distributions obey conditional-independence constraints. It applies the method to bilocal and more complex networks, including examples that distinguish local from bilocal correlations and quantify the correlation needed for classical reproduction.

II. POLYNOMIAL BELL INEQUALITIES

  • The derivation applies to DAGs displaying conditional independencies at the level of the joint distribution, and the paper illustrates it in bilocality.The bilocal example involves three parties whose correlations are mediated by two independent sources.Source: §II · p. 3
  • The method combines linear probability constraints with nonlinear conditional-independence constraints, eliminating non-observable variables to obtain polynomial inequalities in observable data.It can fully characterize the considered GLHV models, though computational tractability may require partial characterizations such as full correlators only.Source: §II · p. 3
  • In the bilocal scenario, Fourier–Motzkin elimination combines the source-independence constraint with linear inequalities to derive a polynomial inequality.The worked derivation combines inequalities in the non-observable correlators and then removes the remaining hidden term.Source: §II · p. 3
  • The construction can quantify relaxed independence: for I = J = 2, classical reproduction requires CAC = 1, or maximal correlation between A and C.The violation of the relaxed inequality quantifies the degree of correlation required to reproduce a non-bilocal correlation classically.Source: §II · pp. 3–4
  • The method derives new polynomial inequalities for the more complicated GLHV model and bilocality; with B0/B1, I=J=9v is nonlocal only for v>5/9 but non-bilocal for v>4/9.Here x,z∈{0,1,2}; the example is quantum-achievable for v≤1/2, while without independence |I|+|J|≤10.Source: §II · p. 4

III. NONSIGNALLING CORRELATIONS AND GENERALIZED BAYESIAN NETWORKS

Nonsignalling (NS) correlations have consistent observable marginals that do not depend on other parties’ measurement choices. In generalized networks, the paper defines generalized nonsignalling (GNS) correlations by replacing classical hidden variables with NS distributions, retaining observable independencies.

Nonsignalling correlations

  • NS correlations require well-defined observable distributions and marginals that do not depend on which measurements other parties perform.The paper represents these conditions as linear inequalities defining a polytope with finitely many extremal points.Source: §III · p. 4

Generalized nonsignalling correlations

  • GNS correlations are compatible with GLHV models in which classical hidden variables are replaced by general NS distributions.The observable conditional independencies valid in the classical setup remain valid under this replacement.Source: §III · p. 4
  • In the bilocality network, observable statistics between parties A and C factorize even when the shared resources are NS distributions.Specifically, p(ai, ck) = p(ai)p(ck).Source: §III · p. 4

Characterizing GNS correlations

  • GNS correlations form a non-convex semi-algebraic set defined by the intersection of linear and polynomial inequalities.Treating some variables in the polynomial constraints as free parameters linearizes the description into a convex set with finitely many extremal points, amenable to standard linear programming.Source: §III · p. 4

Example

  • With the A and C marginals fixed at 1/2, one extremal GNS point is p(a,b,c|x,y,z) = 1/4δa⊕b⊕c,y(x⊕z), achievable by replacing the hidden variables with PR-boxes.The example uses the bilocality model with two dichotomic measurements per party.Source: §III · p. 4

IV. DISCUSSION

The discussion presents the method and its applications to several GLHV models as a step toward deriving polynomial Bell inequalities in generalized Bayesian networks. It also identifies future applications and scenarios for investigation.

Contribution

  • The paper presents a practical, general method applied to several GLHV models to derive polynomial Bell inequalities characterizing them.The authors describe this as an important step in generalizing Bell’s theorem to more complex causal structures.Source: §IV · p. 4
  • The construction also yields a notion of nonsignalling correlations for GLHV models.The discussion presents this as a further consequence of the method.Source: §IV · p. 5

Future directions

  • The authors identify applying the method across varied scenarios and deriving inequalities suited to reducing experimental requirements as next steps.They also propose investigating polynomial Bell inequalities in quantum cryptography, randomness generation, and distributed computing.Source: §IV · p. 5

Appendix A: A method for the derivation of polynomial Bell inequalities

The appendix describes an adapted Fourier–Motzkin elimination procedure for deriving polynomial Bell inequalities from probability and conditional-independence constraints. It explains how free parameters linearize nonlinear constraints and notes computational choices and scope.

Elimination procedure

  • The method combines probability constraints Lp ≥ 0 with polynomial conditional-independence constraints Wp ≥ 0, then eliminates non-observable quantities to obtain inequalities in observable data.Some DAGs, including the triangle scenario, have no conditional independencies at the level of p.Source: Appendix A · p. 6
  • Components appearing nonlinearly in the constraints are retained as free real parameters, while other non-observable terms can be eliminated by ordinary Fourier–Motzkin elimination.The procedure eliminates a term by combining valid inequalities with opposite signs for that term; the remaining terms use the adapted method.Source: Appendix A · p. 6

Computational choices

  • When the observable-term set is small, listing extremal points in the reduced subspace and dualizing may be faster than ordinary Fourier–Motzkin elimination.The appendix notes that the better approach depends on the scenario.Source: Appendix A · p. 6

Bilocality example

  • For the bilocal example with two dichotomic observables per party, the appendix uses a 64-component correlation vector and linearizes the bilocality constraints with free parameters for probabilities at A.The bilocality constraints do not depend on B’s measurement variables, so terms involving them can first be eliminated by ordinary Fourier–Motzkin elimination.Source: Appendix A · p. 6

Scope and final elimination

  • For computational reasons, the appendix’s examples use full correlators without marginal terms, although the method can also yield a full GLHV characterization including marginals.After elimination, observable constraints on the free parameters can be used with quantifier elimination to produce polynomial inequalities involving observable data only.Source: Appendix A · p. 7

Appendix B: Detailed derivation of the polynomial Bell inequalities

The appendix derives polynomial inequalities for bilocal and related GLHV scenarios by combining source-independence constraints with linear correlator constraints and eliminating unobservable terms. It establishes a full-correlator facet result for one bilocal setting and extends derivations to other settings.

Bilocality with two observables

  • For bilocality with two dichotomic observables per party, the derived inequality is a facet of the bilocal set in the full-correlator subspace.The same analysis applies when party B instead performs one four-outcome measurement, such as a Bell-basis measurement.Source: Appendix B · p. 7

Bilocality derivation

  • The derivation can use a simple subset of the full bilocal constraints, then minimize over the unobservable correlator to obtain an inequality in observable data.The appendix describes the intermediate constraint as convex in ⟨A0A1⟩, with its minimum at ⟨A0A1⟩ = (±I ∓ J)/4.Source: Appendix B · p. 7

Correlated A and C

  • Allowing correlation between A and C relaxes the source-independence relation by a bound CAC, yielding modified inequalities.The appendix obtains these by combining the bounded-deviation condition with the earlier bilocal constraints.Source: Appendix B · p. 8

Other GLHV scenarios

  • The appendix also derives analogous inequalities for the four-party GLHV model and for a bilocal scenario with three observables at A and C.For the four-party case it uses independence constraints and linear inequalities; for the three-observable case it combines constraints to eliminate correlators before minimizing over a remaining non-observable quantity.Source: Appendix B · pp. 7–8
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