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No-signalling-projection-invariant Bell inequalities

Soumyadip Patra, Jitendra Prakash, Aaditya Paudel, Peter Bierhorst

TL;DR

The paper addresses the problem of finite-sample weak signalling in Bell-test data by proposing a no-signalling (NS) affine-hull projection using an exact, closed-form L2 projection. It shows that full correlators and uniformly-averaged marginal correlators lie in the NS constraint kernel, enabling a simple three-map construction that projects an empirical frequency onto the NS affine hull without changing the Bell value in a canonical correlator form. This yields projection-invariant canonical Bell expressions, including tilted CHSH, I3322, and LOSR-GTNL inequalities, and extends naturally to weighted L2 projections to accommodate nonuniform setting distributions. The practical impact is a robust, efficient preprocessing step that standardises Bell-value comparisons across experiments and streamlines device-independent tasks such as randomness generation, QKD, and entanglement certification by isolating genuine nonlocality from signalling artefacts.

Abstract

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 $L^2$-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.

No-signalling-projection-invariant Bell inequalities

TL;DR

The paper addresses the problem of finite-sample weak signalling in Bell-test data by proposing a no-signalling (NS) affine-hull projection using an exact, closed-form L2 projection. It shows that full correlators and uniformly-averaged marginal correlators lie in the NS constraint kernel, enabling a simple three-map construction that projects an empirical frequency onto the NS affine hull without changing the Bell value in a canonical correlator form. This yields projection-invariant canonical Bell expressions, including tilted CHSH, I3322, and LOSR-GTNL inequalities, and extends naturally to weighted L2 projections to accommodate nonuniform setting distributions. The practical impact is a robust, efficient preprocessing step that standardises Bell-value comparisons across experiments and streamlines device-independent tasks such as randomness generation, QKD, and entanglement certification by isolating genuine nonlocality from signalling artefacts.

Abstract

In this paper, we highlight how any Bell inequality for a configuration involving parties each performing one of binary-outcome measurements has a canonical form that is no-signalling-projection invariant. Specifically, the -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.

Paper Structure

This paper contains 16 sections, 3 theorems, 69 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Projection in the nm2 Bell scenario
  3. The affine subspace AcontainsN
  4. Two approaches to no-signalling approximation: ML and L2-projection
  5. Row structure and sparsity of Aeq
  6. Uniformly-averaged marginal correlators
  7. Closed formula for projection
  8. Canonical Bell expressions
  9. Family of tilted CHSH inequalities
  10. I3322 inequality
  11. Bell-like inequality for LOSR-GTNL
  12. Generalisation to weighted L2 norms
  13. Conclusion
  14. Acknowledgements
  15. Proof of Proposition \ref{['prop:CorrVecBelongsToNullSpace']}
  16. ...and 1 more sections

Key Result

Proposition 1

For all $\tilde{\mathbf{u}}_{I}\in\mathcal{S}^{k}$, the UMC coefficient vector $\mathbf{c}^{I}_{\tilde{\mathbf{u}}_{I}}$ satisfies $\mathbf{c}^{I}_{\tilde{\mathbf{u}}_{I}}\in\mathrm{ker}(A_{\mathrm{eq}})$.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Corollary 2: Invariance of k-party UMC under Pi_A
  • proof
  • Proposition
  • proof