No-signalling-projection-invariant Bell inequalities

The gist The canonical form for Bell expressions can be derived using uniformly-averaged marginal correlators, which makes them termwise invariant under projection onto the no-signalling affine hull, thereby isolating nonlocality from weak signalling artifacts in experimental data

How it works

The paper highlights how any Bell inequality for a configuration involving n parties each performing one of m binary-outcome measurements has a canonical form that is no-signalling-projection invariant, specifically stating that the L2-projection of weakly signalling data onto the no-signalling polytope leaves the violation of this canonical Bell inequality unchanged The methods allow for a general closed formula for the projection and a computationally simpler procedure for its evaluation, which can be generalized to non-standard projections This projection serves as a preliminary step before undertaking any device-independent application involving Bell experiment data, such as hypothesis testing against local realism, random number generation and entanglement detection

Two approaches to no-signalling approximation

The paper compares the maximum-likelihood (ML) method with the L2-projection method for obtaining a no-signalling approximation of an empirical frequency vector f The ML estimator is obtained by maximizing the log-likelihood over the no-signalling set PNS, which involves solving a convex program that can become taxing as the number of parties n and measurement settings size m grow In contrast, the L2-method is a closed-form affine (orthogonal) projection onto A, which is described by the formula pb = ΠA(f):= Πker(f − d) + d This approach is faster, stabler, and more interpretable for analysis

Projection in the (n, m, 2) Bell scenario

The mathematical setting involves n non-communicating parties sharing a quantum-entangled resource and obtaining binary outcomes based on their input settings A behaviour p is geometrically viewed as a vector in R d whose components are the settings-conditional outcome probabilities, and it belongs to the orthant R d+ satisfying normalization conditions The set of no-signalling conditions, along with normalisation conditions, defines an affine hull A which is the affine hull of PNS

Uniformly-averaged marginal correlators

The paper formalizes a canonical, symmetry-respecting choice of functional in p that is linear in p and invariant under permutation of the settings choices outside the k-party set This quantity, referred to as the k-party uniformly-averaged marginal correlator (UMC), is defined as C¯I xI(p):= 1/(mn-I) X xIbar X a χI (a)p(ax) A key result is that the UMC coefficient vectors c Iu I satisfy c Iu i ∈ ker(Aeq), and these vectors span the entire subspace ker(Aeq)

Closed formula for projection

The paper derives a simplified, closed-form approach to constructing the projector onto A by first deriving an inverse relation that expresses a no-signalling behaviour in terms of its correlators The L2-projector sending any weakly-signalling behaviour f to its projection onto A is given by pb(ax) = 1/2 n X I⊆[n] χI (a) 1/(mn-I) X xIbar a' χI (a')f(a'x) This formula can be decomposed into a sequence of three simple linear maps T3T2T1f, which allows for structured sparsity and avoids dense inverses

Canonical Bell expressions

The kernel membership of the k-party UMC coefficient vectors shows that L2-projection onto A preserves all correlator values, leading to a class of robust Bell inequalities whose evaluation is stable under projection of empirical behaviours onto the no-signalling affine hull Specifically, there exist different coefficients such that B(ΠA(p)) = B(p) for a linear Bell expression in the canonical correlator form This canonical form can be taken as a starting point that minimises leverage from signalling

Generalisation to weighted L2 norms

The work on L2 projections is extended to scenarios where settings occur with different probabilities by encoding the different weightings with a generalised inner product ⟨·, ·⟩D This leads to the weighted projection pb = arg min p∈A f - p D squared The invariance of averaged correlators under this weighted projection means that the results of Section 3 can be immediately adapted to generate a simple closed-form expression for the D-weighted projection

Conclusion

The canonical, project-then-evaluate workflow standardises comparisons across experiments and benefits device-independent tasks such as randomness generation, QKD, and multipartite nonlocality/entanglement certification by ensuring that the Bell value reflects nonlocal structure rather than residual signalling The three map pipeline implements the projection with structured sparsity, avoiding dense inverses and scaling cleanly to larger instances This canonical workflow standardises comparisons across experiments and benefits device-independent tasks such as randomness generation, QKD, and multipartite nonlocality/entanglement certification by ensuring that the Bell value reflects nonlocal structure rather than residual signalling The canonical form isolates the violation attributable to strictly no-signalling nonlocality, providing a cleaner and more comparable metric of nonlocal behaviour across experiments

How it works

The probability bound for the event of negative entries in pb is derived using Hoeffding/McDiarmid inequality, which provides an upper bound on the probability that a projected coordinate becomes negative This bound shows that P[pb(ax) < 0] ≤ exp − 2p0(ax) squared / P x' ∆a,x(x') 2/Nx' The calculation for the balanced case in (33) yields an expression where P x' ∆a,x(x') 2/Nx' = Cn,m/N with Cn,m = 4m 2/(2m) 2n(m+1) squared [mn/(m + 3)n − (m − 1)n] This bound shows that for a (2,2,2) Bell experiment with all underlying outcome probabilities at least 0.05—a reasonable assumption for an experiment aiming to observe CHSH values close to the Tsirelson bound—the probability of observing negative probabilities is already near-negligible (< 3 × 10−4) with N = 1, 500 trials for each setting

How it works

The matrix representation of the linear map T1 has a block structure where the Walsh-Hadamard matrix H4 in R 4x4 along the diagonal is expressible as H4 = SH⊗2 for H = 1 1 / 1-1!

1. textbfRewriting Bell Inequality Evaluation for Robustness: Canonical Form Usage (Section 4 and 6). Improving AI systems can integrate a canonical form step before evaluating Bell inequalities, as the paper states: The canonical form is taken as its starting point that minimises leverage from signalling. This ensures that weakly-signalling artifacts from finite data do not affect the perceived nonlocality as indicated by a Bell inequality violation, leading to more stable and interpretable results in device-independent tasks like entanglement certification.

2. textbfImplementing L2-Projection for Empirical Data Estimation (Section 3). AI systems can use the derived closed formula for the projection: pb(ax) = 1/2 n X I⊆[n] χ I (a) 1/(mn-I) X xˉ,a′ χ I (a′) f(a'x), to replace standard Maximum Likelihood estimates. This projection attends the L2-projection of an empirical frequency onto A, which is faster, stabler, and more interpretable for our analysis compared to iterative ML routines.

3. textbfHandling Non-Uniform Sampling in Weighted Projections (Section 5). AI systems can adapt the projection method to non-uniform settings distributions by using a generalized inner product: ⟨p1, p2⟩D = p T 1 D p2, where D is a diagonal matrix reflecting setting weights. This allows the system to perform the weighted projection of f onto the no-signalling space using the formula pb = arg min p∈A f - p D, which is crucial for protocols like device-independent quantum key distribution where settings are sampled non-uniformly.

4. textbfProviding Finite-Sample Negative Probability Bounds (Section 3.1). When projecting empirical data, AI systems can use the derived Hoeffding/McDiarmid inequality to quantify risk: "P[pb(ax) < 0] ≤ exp − 2p0(ax) squared / P x' ∆a,x(x') squared / Nx'," which provides a concrete bound on the probability that the projected behavior has negative coordinates, helping to rule out issues in experimental implementations.

5. textbfSimplifying Computational Pipelines via Sparse Matrix Decomposition (Section 3). The AI can implement the projection using structured sparsity by decomposing the map into three linear maps, showing the composition of three simple linear maps T3T2T1f, which avoids dense inverses and scaling cleanly to larger instances compared to direct projector construction.

In this paper, we highlight how any Bell inequality for a configuration involving n parties each performing one of m binary-outcome measurements has a canonical form that is no-signalling-projection invariant. Specifically, the L squared-projection of weakly signalling data onto the no-signalling polytope leaves the violation of this canonical Bell inequality unchanged. Our methods allow us to derive a general closed formula for the projection and present a substantially more computationally simple procedure for its evaluation. We also show this can be generalised to non-standard projections of potential interest for certain applications. No-signalling projections serve as a preliminary step before undertaking any device-independent application involving Bell experiment data, such as hypothesis testing against local realism, random number generation and entanglement detection.