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Genuine quantum nonlocality in the triangle network

Marc-Olivier Renou, Elisa Bäumer, Sadra Boreiri, Nicolas Brunner, Nicolas Gisin, Salman Beigi

arXiv:1905.04902v1quant-ph

TL;DR

The paper asks whether no-input quantum nonlocality can be genuinely network-specific, beyond examples traceable to Bell-inequality violations. It constructs triangle-network quantum distributions using entangled sources and joint entangled measurements, and reports examples not reproducible by trilocal models, with generalizations to odd-cycle networks.

  • Problem

    Earlier examples of quantum nonlocality without inputs could be traced to standard Bell-inequality violations, while independent sources make the set of triangle-network distributions non-convex and difficult to characterize.

    Source: §I · p. 1, §II · p. 2

  • Method

    The authors construct quantum distributions using entangled states from independent sources and joint entangled measurements, then test whether trilocal models can reproduce them.

    Source: §II · p. 2

  • Results

    The paper presents triangle-network quantum distributions that cannot be reproduced by classical trilocal models and generalizes its construction to odd-cycle networks.

    Source: §II · p. 2, §I · p. 2

  • Takeaways & Limitations

    The authors present these examples as quantum nonlocality genuine to network configurations rather than a consequence of standard Bell nonlocality.

    Source: §IV · p. 4

    The first computationally implementable inflation tests appear unable to detect even the nonlocality of the constructed distribution, although the method is known to converge in the limit.

    Source: §IV · p. 4

Abstract

from arXiv · show

Quantum networks allow in principle for completely novel forms of quantum correlations. In particular, quantum nonlocality can be demonstrated here without the need of having various input settings, but only by considering the joint statistics of fixed local measurement outputs. However, previous examples of this intriguing phenomenon all appear to stem directly from the usual form of quantum nonlocality, namely via the violation of a standard Bell inequality. Here we present novel examples of 'quantum nonlocality without inputs', which we believe represent a new form of quantum nonlocality, genuine to networks. Our simplest examples, for the triangle network, involve both entangled states and joint entangled measurements. A generalization to any odd-cycle network is also presented. Finally, we conclude with some open questions.

I. INTRODUCTION

The introduction frames quantum networks as a setting for fixed-output quantum nonlocality, then asks whether examples can go beyond phenomena traceable to ordinary Bell-inequality violations. It presents a triangle-network construction using entangled states and joint entangled measurements, and reports generalizations to odd-cycle networks.

I. INTRODUCTION

  • Bell’s theorem is described as foundational to quantum theory, influential in quantum information science, and a basis for the device-independent paradigm.The introduction notes that loophole-free tests of quantum nonlocality were reported and provided a basis for device-independent quantum information protocols.Source: §I · p. 1
  • The standard multipartite Bell approach considers three or more observers sharing an entangled state distributed by a common source.The introduction distinguishes this common-source setup from quantum networks, where multiple independent sources distribute entanglement.Source: §I · p. 1
  • Quantum networks permit nonlocality tests with fixed local measurements and no inputs, unlike standard approaches that use measurement settings.Network sources are assumed independent, and joint entangled measurements can correlate systems across the network.Source: §I · p. 1
  • Earlier examples of quantum nonlocality without inputs could be traced to standard Bell-inequality violations, motivating the search for a network-genuine form.The authors ask whether network configurations can yield a genuinely novel form of quantum nonlocality.Source: §I · p. 1
  • The proposed triangle-network examples combine entangled states with joint entangled measurements, and the authors argue they differ fundamentally from previously known forms of quantum nonlocality.The paper also presents generalizations to cycle networks with an odd number of parties.Source: §I · p. 2

II. SCENARIO AND MAIN RESULT

The paper constructs fixed-output quantum correlations for the triangle network and proves that, over a specified measurement-parameter range, no classical model with independent sources can reproduce them. The proof derives necessary constraints on any such model and shows they cannot all hold.

II. SCENARIO AND MAIN RESULT

  • The triangle has three pairwise sources, no shared tripartite information, and no measurement inputs; its observed data are the joint outputs P(a, b, c).Characterizing its classical distributions is difficult because source independence makes the set non-convex, and strong classical bounds are missing.Source: §II · p. 2
  • The proof uses a logical contradiction rather than a Bell-inequality violation, deriving necessary properties of any trilocal model and showing they cannot all hold together.The proof includes forbidden joint outputs and a constraint that the number of parties outputting χ is odd.Source: §II · pp. 2–3
  • Each of three independent sources distributes a maximally entangled two-qubit state, and each observer performs the same projective measurement on two received subsystems.The measurement basis depends on a real parameter and includes two entangled states.Source: §II · p. 2
  • For u_max^2 < u^2 < 1, the constructed quantum distribution cannot be reproduced by any classical trilocal model.The lower threshold is approximately 0.785.Source: §II · pp. 2–3
  • A trilocal model must partition each source's classical variable set into two equal-weight subsets that determine coarse-grained outputs.The proof then defines a distribution over the χ outcomes and partition labels; its required marginals cannot be jointly compatible in the nonlocality range.Source: §II · p. 3, Figure 2 · p. 3
  • An explicit trilocal model exists at u^2 = u_max^2, and the authors conjecture trilocality extends below that threshold.They also note a construction for u^2 = 1/2 using a uniformly random choice between χ0 and χ1.Source: §II · p. 3

III. GENERALISATIONS

The paper extends triangle-network nonlocality results to generalized two-qubit source states, qutrit systems, and odd-cycle networks. The odd-cycle result applies asymptotically as the measurement parameter approaches 1, but the approach does not extend directly to even cycles.

Generalized source states

  • For general pure two-qubit source states, nonlocality holds for umax(λ0) ≲ u < 1; the lowest threshold occurs at λ0 = 2/3, requiring a non-maximally entangled joint measurement.The sources produce states λ0 |00⟩+ λ1 |11⟩, with λ0² + λ1² = 1.Source: §III · p. 3

Higher-dimensional systems

  • With maximally entangled two-qutrit sources and nine-outcome joint projective measurements, suitable parameters yield a distribution incompatible with any trilocal model.Source: §III · pp. 3–4

Odd-cycle networks

  • For any odd N-cycle, maximally entangled two-qubit sources and the same joint measurement yield incompatibility with N-local models as u approaches 1; the approach does not directly extend to even cycles.Source: §III · p. 4

IV. DISCUSSION

The discussion contrasts the proposed network nonlocality with Fritz’s construction, which embeds a standard Bell test in the triangle. It also identifies limits in proving measurement requirements, establishing noise tolerance, and quantifying randomness.

Distinction from Fritz’s construction

  • Unlike Fritz’s example, which embeds a CHSH test using effective inputs from the other sources, the authors present their construction as not stemming from standard Bell nonlocality.Source: §IV · p. 4
  • The construction is symmetric, with four outputs per party, while symmetrizing Fritz’s example requires additional outputs; the authors also believe PR-box simulation is unavailable.They attribute that belief to the use of joint measurements with entangled eigenstates, unlike Fritz’s separable measurements.Source: §IV · p. 4

Open questions

  • The claim that the distribution requires all sources to be entangled and joint entangled measurements remains qualitative, without a formal proof.Source: §IV · p. 4

Noise tolerance

  • The main theorem concerns the exact noiseless distribution; although some noise robustness follows, the authors have not obtained reasonable noise tolerance.Source: §IV · p. 4
  • The authors propose inflation as a possible route to a nonlinear Bell inequality, but its first computer-implementable tests appear unable to detect this distribution’s nonlocality.They note that inflation is known to converge in the limit.Source: §IV · p. 4

Randomness

  • Device-independent quantification of the distribution’s likely intrinsic randomness remains an open question under the assumption of source independence.Source: §IV · p. 4

Appendix A: Proof of main result

The proof assumes a trilocal model, reduces its possible outputs to constraints on independent-source subsets, and shows those constraints conflict with quantum marginals for a range of measurement parameters.

Generalization of Theorem 1

  • The proof extends the result to shared non-maximally entangled two-qubit pure states, with the threshold depending on their Schmidt coefficients.The shared states are |ψ⟩ = λ₀|00⟩ + λ₁|11⟩.Source: Appendix A · pp. 5–6

Proof of Theorem 1

  • Coarse-graining χ₀ and χ₁ into χ forces a trilocal model to partition each source space into disjoint subsets that determine the outputs.The subsets have probabilities λ₀² and λ₁², respectively.Source: Appendix A · pp. 6–7
  • The proof then groups outputs and hidden-variable subsets into a probability distribution whose required marginals are incompatible for a suitable measurement basis.This contradiction does not rely on violating a Bell-type inequality.Source: Appendix A · pp. 6–7

Theorem 1

  • For suitable u, the quantum distribution has no trilocal model when u² lies above the λ₀-dependent threshold and below 1.The threshold u_max(λ₀) is implicitly defined by a constraint; the proof obtains the contradiction from incompatible marginals.Source: Appendix A · p. 7

Appendix B: Generalisation to qutrits

The qutrit generalization uses two-qutrit maximally entangled states and joint measurements to derive constraints on any trilocal model. For a suitable orthonormal measurement basis, those constraints imply that no such model reproduces the quantum distribution.

Qutrit construction

  • The construction uses shared two-qutrit maximally entangled states and joint measurements with nine outcomes.The outcomes are labeled by three ordinary outcomes and χ-type outcomes.Source: Appendix B · p. 9

Proof of Theorem 2

  • The proof derives marginal constraints for a grouped output-and-hidden-variable distribution, then tests their compatibility as a linear programming problem.The paper exhibits an orthonormal-basis example for which the required distribution has no solution.Source: Appendix B · pp. 10–12
  • A coarse-grained trilocal model must partition each source space into three disjoint, equally weighted subsets that determine the parties’ outputs.Each of the subsets has probability 1/3.Source: Appendix B · pp. 10–11

Theorem 2

  • Theorem 2 states that for some choice, the quantum distribution PQ(a, b, c) is incompatible with any trilocal model.The argument implies q(i = 1, j = 0, k = 1) = −1/30, which is negative.Source: Appendix B · pp. 10, 12

Appendix C: Generalisation to odd-cycle networks

The appendix generalizes the triangle argument to odd-cycle networks, with maximally entangled qubit states and the same joint measurement at every party. It proves that for u asymptotically close to 1, the resulting distribution has no N-local model.

Odd-cycle construction

  • The generalization considers N parties connected in a cycle by maximally entangled qubit states, with each party using the same projective measurement basis.The measurement basis is the one used in the main-text construction.Source: Appendix C · p. 12

Theorem 3

  • For odd N and u asymptotically close to 1, no N-cycle classical model reproduces the quantum distribution.The proof reaches a contradiction from the constraints required of a local hidden-variable model.Source: Appendix C · pp. 13, 14, 16

Proof of Theorem 3

  • As in the triangle proof, grouping χ₀ and χ₁ into χ constrains source-set memberships, which are used to derive probability-distribution marginals.The proof shows those marginals are incompatible for the specified parameter regime.Source: Appendix C · pp. 13–14

Appendix D: Trilocal model

This appendix constructs a trilocal model for the maximally entangled triangle distribution at the measurement parameter saturating the main theorem’s bound. The construction satisfies its required constraints using numerical verification for one branch and analytic verification for the other.

Trilocal model

  • At the measurement parameter saturating the bound, the appendix constructs a trilocal strategy that reproduces the quantum distribution.The case considered uses maximally entangled states.Source: Appendix D · p. 16

Strategy construction

  • The construction uses symmetric strategies for two branches, with local variables taking values in three-element or two-element sets and outputs determined by ordered pairs.The strategy specifies χ₁ for the t = 0 branch and χ₀ for the t = 1 branch under the stated pair conditions.Source: Appendix D · p. 16

Verification

  • The required constraints are supported numerically for t = 0 and verified analytically for t = 1.The authors report that they could not analytically simplify the t = 0 expressions.Source: Appendix D · p. 16
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