Measuring Bell non-locality in the presence of signaling

Abstract

Scientific inquiry seeks causal explanations of observed phenomena. The Bell experiment provides a paradigmatic case, revealing correlations between spatially separated systems that no local model can reproduce. Such correlations, known as Bell non-locality, are typically analyzed under the non-signaling assumption, which requires that local statistics be independent of distant measurement choices. Yet real experiments, as well as applications beyond physics, often involve signaling, raising the question of how non-locality should be characterized without this constraint. We introduce a general method for quantifying Bell non-locality in the presence of signaling, designed to relax locality as little as necessary. Our approach is guided by the question: how often can locality be preserved across repeated trials in explaining the observed correlations? The task reduces to finding the optimal convex decomposition of the observed correlations into local and genuinely non-local components. We solve this problem within the linear-programming framework, obtaining a closed-form solution valid for arbitrary correlations. We further evaluate a corresponding measure of signaling, demonstrating the generality of the method and the non-trivial character of the results. By extending the notion of non-locality beyond the non-signaling regime, our framework reshapes the basis for experimental analysis in physics and offers tools applicable outside physics.

Figures (8)

• Figure 1: Summary of the results. The fractional measure reflects the proportion of experimental trials that can still be understood within a specific explanatory framework, be it local or non-signaling. We calculate it explicitly and show that the result takes the form of a minimum over simpler linear measures, each represented by a vector that projects the observed statistics onto the relevant directions of the correlation space. Accordingly, the table presents the collections of vectors for each measure, with 128 vectors specifying the local fraction (Theorem \ref{['theoremL']}) and 120 vectors specifying the non-signaling fraction (Theorem \ref{['theoremNS']}). Importantly, none of these vectors can be omitted and some coincide with familiar quantities: notably, the f-vectors correspond to maximal signaling (Lemma \ref{['Lemma-muNS-f']}), while the e-vectors align with the Bell-CHSH expressions (Theorem \ref{['Theorem-muL-e']}).
• Figure 2: Causal explanations of Bell experiment. Two parties, Alice and Bob, perform experiments at spatially separated locations. For each freely chosen pair of measurement settings $x$ and $y$, they obtain outcomes $a$ and $b$, respectively, giving rise to the observed statistics $\{P_{ ab|xy}\}_{ xy}$. Assuming that causes propagate only forward-in-time, such statistics can be described by directed acyclic graphs (DAGs), where nodes represent measurement settings, outcomes, or shared variables, and arrows capture potential cause-and-effect relations. On the left, the causal DAG depicts a situation where the locality condition rules out any direct causal influence between the parties, so correlations can only originate from a shared common cause in the past, denoted by $\lambda$. On the right, the presence of any red arrow indicates a direct causal influence and therefore constitutes a violation of the locality assumption. While the local scenario cannot reproduce arbitrary correlations ${P_{ ab|xy}}$, allowing red arrows in the non-local scenario---potentially active in every experimental trial---can generate any behavior. In this work, we investigate to what extent the correlations observed in Bell experiments can be accounted for by minimally relaxing locality, that is, by introducing non-local (red) arrows as sparingly as possible across repeated trials.
• Figure 3: Geometry of behaviors. Schematic illustration of the nested structure of convex polytopes embedded in $\mathbb{R}^{ 16}$. Each relaxation of constraints enlarges the set: from the local polytope$\mathcal{P}_{ Loc}$ (8-dimensions) to the non-signaling polytope$\mathcal{P}_{ NS}$ (8-dimensions), and finally to the full correlation space (polytope)$\mathcal{P}$ (12-dimensions).
• Figure 4: Significance of the vector measures. Numerical results showing the proportion of randomly sampled behaviors for which the solutions in Eqs. (\ref{['muL-solution']}) and (\ref{['muNS-solution']}) are uniquely attained by each vector type within the respective subsets of $\mathbb{Q}$ (green) and $\mathbb{S}$ (yellow). Evidently, the f-measures (maximal signaling) dominate for most behaviors, although a considerable fraction is accounted for by the remaining ones. Interestingly, for a random behavior, the e-measures (Bell-CHSH expressions) are the least likely to contribute to the quantification of locality, while various forms of signaling corrections (f-, g-, t-, s-, and z-measures) play a more prominent role.
• Figure S2: 'Alternating' direction of signaling violation. 1) There are two parties, A and B, with two arrows from A to B (4,3) and two arrows from B to A (2,1). 2) We traverse a path starting from B or A using each arrow only once. 3) When traversing an arrow we carry a flag L or R (these may differ along a composite path). 4) Path reversals are equivalent, e.g. L along 1 then R along 3 is equivalent to R along 3 then L along 1. 5) Each path has a unique start and end.

Theorems & Definitions (13)

• Definition 1: Measure of locality
• Definition 2: Measure of non-signaling
• Theorem 1
• Lemma 1
• Theorem 2
• Theorem 3
• proof : Justification
• proof : whiteJustification
• proof : whiteProof of Lemma \ref{['Lemma-muNS-f']}
• proof : whiteProof of Theorem \ref{['Theorem-muL-e']}