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Nonlinear Bell inequalities tailored for quantum networks
Denis Rosset, Cyril Branciard, Tomer Jack Barnea, Gilles Pütz, Nicolas Brunner, Nicolas Gisin
TL;DR
Characterizing correlations from independent sources is challenging because source independence makes the set of N-local correlations non-convex. The paper iteratively constructs Bell inequalities for a larger network by adding one source and one observer to a network with an existing inequality. The examples show strong quantum violations; the quantum correlations obtained violate Bell inequality (11), and are non-N-local, for V > 2−N/2.
Problem
Characterizing correlations from independent sources is challenging because source independence makes the set of N-local correlations non-convex.
Source: p. 2Method
The paper iteratively constructs Bell inequalities for a larger network by adding one source and one observer to a network with an existing inequality.
Source: p. 1Results
The examples show strong quantum violations; the quantum correlations obtained violate Bell inequality (11), and are non-N-local, for V > 2−N/2.
Source: p. 4Takeaways & Limitations
The method applies iteratively to a broad class of networks and provides inequalities whose violation certifies that correlations have no N-local model.
Source: p. 2The method adds leaves but does not create cycles, so the paper treats only acyclic networks.
Source: p. 5
Abstract
from arXiv · showhide
In a quantum network, distant observers sharing physical resources emitted by independent sources can establish strong correlations, which defy any classical explanation in terms of local variables. We discuss the characterization of nonlocal correlations in such a situation, when compared to those that can be generated in networks distributing independent local variables. We present an iterative procedure for constructing Bell inequalities tailored for networks: starting from a given network, and a corresponding Bell inequality, our technique provides new Bell inequalities for a more complex network, involving one additional source and one additional observer. The relevance of our method is illustrated on a variety of networks, for which we demonstrate significant quantum violations.
Appendix A: General statement and proof of our main theorem
Appendix A extends the iterative construction to more general Bell constraints and proves that the resulting inequalities hold for N′-local correlations. The extension includes non-full-correlation terms and can use a shared quantifier value across multiple initial inequalities.
Appendix A: General statement and proof of our main theorem
- Adding a source linked to an existing party and a new observer yields novel constraints for correlations in the enlarged network.Source: Appendix A · p. 5
- Theorem 2 generalizes Theorem 1 to inequalities with non-full-correlation terms and allows the same q for multiple initial inequalities.Source: Appendix A · p. 6
- The proof handles the new source's independent variable by splitting an existing party's inputs into two subsets and averaging over the new variable.Source: Appendix A · p. 6
- The theorem's input constraints may already contain quantifiers, so the construction can be applied iteratively.Source: Appendix A · p. 6
Appendix B: Obtaining CHSH from the proof of Theorem 2
Appendix B uses the proof of Theorem 2 to derive the familiar CHSH inequality from a one-party network.
Appendix B: Obtaining CHSH from the proof of Theorem 2
- Starting with a network containing only A1, adding a source and A2 gives the Bell-local scenario, whose correlators satisfy CHSH.Source: Appendix B · p. 7
- The derivation follows the averaging step in the proof, where a convex sum of quantities in [−1, 1] yields the required bound.Source: Appendix B · p. 7
Appendix C: Recovering the bilocal inequality of Ref. [9]
Appendix C shows that applying the construction to equivalent CHSH inequalities recovers the bilocal inequality of Ref..
Appendix C: Recovering the bilocal inequality of Ref. [9]
- Combining four inequalities obtained from equivalent CHSH forms yields the bilocal inequality derived in.Source: Appendix C · p. 7
Appendix D: Bell inequality for trilocality in the nonlinear form
Appendix D removes quantifiers from the trilocality inequality to obtain a nonlinear form. The resulting polynomial is cumbersome, but the stated quantum correlations violate it above the specified visibility threshold.
Appendix D: Bell inequality for trilocality in the nonlinear form
- The nonlinear form for the three-source, four-observer chain is obtained by rewriting Eq. (10) without quantifiers.Source: Appendix D · p. 7
- The nonlinear form loses discriminating power when any of I, J, K, or L equals zero.Source: Appendix D · p. 7
- Using an elimination ideal to remove q and r gives polynomial W of degree 9 with 286 terms, which can be decomposed using primary and secondary invariants.Source: Appendix D · p. 7
- For the stated quantum correlations, the polynomial inequality is violated when V > 2^(-3/2).Source: Appendix D · p. 8
Appendix E: Recovering the Bell inequalities of Ref. [10] for star networks
Starting from all four CHSH versions gives a stronger constraint than Eq. (15). Minimizing the left-hand side over the q parameters yields a constraint equivalent to the star-network Bell inequality derived in.
Appendix E: Recovering the Bell inequalities of Ref. [10] for star networks
- Starting from all four CHSH versions yields a stronger constraint than Eq. (15), expressed as Eq. (E1).Source: Appendix E · p. 8
- Minimizing Eq. (E1)'s left-hand side over q1,...,qN−1 makes it equivalent to the following constraint.Source: Appendix E · p. 8
- The resulting constraint is the Bell inequality for N-local correlations in a star-shaped network derived in.Source: Appendix E · p. 8
Appendix F: Extending the Bell inequality I3322 to bilocality
The appendix applies the extension method to I3322, adding an independent source and observer to obtain a bilocality inequality. It also describes a nonlinear form and alternative inequalities from different choices.
Appendix F: Extending the Bell inequality I3322 to bilocality
- The example starts from I3322 in correlation form, a Bell inequality with more than two inputs and marginal terms.Source: Appendix F · p. 8
- The construction adds observer A3 and independent source S2 connected to A2 and A3, producing the bilocal scenario.Source: Appendix F · p. 8
- Choosing X2 in the extension of I3322 yields an inequality satisfied by bilocal correlations.Source: Appendix F · p. 8
- After adding absolute values, the inequality can be rewritten without a quantifier in nonlinear form.Source: Appendix F · p. 8
- Other choices for X2 or exchanging A1 and A2 yield different bilocal Bell inequalities.Source: Appendix F · p. 8