Measuring Bell non-locality in the presence of signaling

1. Problem Statement

The paper addresses a fundamental limitation in the analysis of Bell non-locality: the standard framework relies heavily on the non-signaling assumption. This assumption posits that the local statistics of one party (e.g., Alice) are independent of the measurement choices made by a distant party (e.g., Bob). While valid for ideal spatially separated quantum systems, this assumption is often violated in:

• Real-world experiments: Due to noise, imperfections, and finite sampling.
• Temporal scenarios: Where measurements occur sequentially, implying causal influence (signaling) is inherent.
• Applications outside physics: Such as cognitive science, economics, and epidemiology, where signaling is often intrinsic to the system design.

Current methods typically treat signaling as a binary "pass/fail" or discard data that violates non-signaling. The authors argue that this is insufficient for quantifying how much non-locality is genuinely present versus how much is an artifact of signaling. The core question is: How can one quantify the degree of Bell non-locality in observed correlations when signaling is present, without making additional assumptions?

2. Methodology

The authors propose a resource-theoretic approach based on convex decomposition and linear programming.

• Fractional Measures: Instead of asking if a behavior is local or non-local, they ask: What is the maximum proportion of experimental trials that can be explained by a local (or non-signaling) model?

  • Local Fraction ($\mu_L$): The maximum weight $p$ such that an observed behavior $\vec{P}$ can be decomposed as $\vec{P} = p \vec{P}_L + (1-p) \vec{P}_{NL}$, where $\vec{P}_L$ is a local behavior and $\vec{P}_{NL}$ is any behavior.
  • Non-Signaling Fraction ($\mu_{NS}$): The maximum weight $p$ such that $\vec{P} = p \vec{P}_{NS} + (1-p) \vec{P}_{S}$, where $\vec{P}_{NS}$ is non-signaling and $\vec{P}_{S}$ is any behavior.

• Geometric Formulation:

  • The set of all possible behaviors forms a 16-dimensional correlation space $\mathcal{P}$.
  • The set of local behaviors forms the local polytope $\mathcal{P}_{Loc}$.
  • The set of non-signaling behaviors forms the non-signaling polytope $\mathcal{P}_{NS}$.
  • The problem is formulated as finding the "distance" (in terms of convex mixing) from an observed behavior $\vec{P}$ to these polytopes.

• Linear Programming (LP) Solution:

  • The authors formulate the maximization of $p$ as a primal Linear Program.
  • They utilize Strong Duality to transform this into a Dual Linear Program.
  • The dual problem reveals that the solution is the minimum of a set of linear scalar products between the observed behavior $\vec{P}$ and a specific set of dual vectors (vertices of the dual polytope).
  • This converts the optimization problem into a closed-form calculation: $\mu(\vec{P}) = \min_{\vec{q} \in \mathcal{Q}} (\vec{q}^T \cdot \vec{P})$.

3. Key Contributions

1. General Framework: The paper provides the first unified, quantitative framework for analyzing Bell non-locality that does not require the non-signaling assumption. It applies to arbitrary correlations, including those with signaling.
2. Closed-Form Solutions: The authors derive explicit, closed-form expressions for both the local fraction ($\mu_L$) and the non-signaling fraction ($\mu_{NS}$) for the standard two-party, two-setting, two-outcome Bell scenario.
3. Vector Characterization:
   • Local Fraction ($\mu_L$): Requires a set of 128 specific vectors ($\mathcal{Q}$) to compute the minimum.
   • Non-Signaling Fraction ($\mu_{NS}$): Requires a set of 120 specific vectors ($\mathcal{S}$).
   • These vectors are categorized into classes (f, g, t, s, z, e) based on the number of non-zero entries and their symmetry properties.
4. Non-Redundancy: The authors prove that none of the 128 or 120 vectors are redundant; for every vector, there exists a specific behavior where that vector uniquely determines the minimum value.

4. Key Results

Theorem 1 (Non-Signaling Case)

For non-signaling behaviors, the local fraction reduces to the known result involving Bell-CHSH inequalities:
$$\mu_L(\vec{P}) = \min \left( 1, \frac{4 - S}{2} \right)$$
where $S$ is the maximum Bell-CHSH value. This confirms the method aligns with standard theory when signaling is absent.

Theorem 2 (General Local Fraction)

For arbitrary behaviors (potentially signaling), the local fraction is:
$$\mu_L(\vec{P}) = \min_{\vec{q} \in \mathcal{Q}} (\vec{q}^T \cdot \vec{P})$$
where $\mathcal{Q}$ contains 128 vectors.

• Structure: The set $\mathcal{Q}$ includes:
  • f-vectors (8): Correspond to maximal signaling signatures.
  • e-vectors (8): Correspond to Bell-CHSH expressions.
  • g, t, s-vectors (112): New terms that capture complex geometric relationships between the local polytope and the full correlation space. These are essential when signaling is present.

Theorem 3 (General Non-Signaling Fraction)

The non-signaling fraction is:
$$\mu_{NS}(\vec{P}) = \min_{\vec{q} \in \mathcal{S}} (\vec{q}^T \cdot \vec{P})$$
where $\mathcal{S}$ contains 120 vectors.

• Structure: Similar to $\mathcal{Q}$ but replaces the s-vectors with z-vectors (where one entry is 2 instead of 1). The e-vectors (Bell-CHSH) are absent from this set, indicating that Bell inequalities alone are insufficient to characterize non-signaling in the presence of signaling.

Geometric Insight

The complexity arises because the solution depends on the embedding of the target polytope ($\mathcal{P}_{Loc}$ or $\mathcal{P}_{NS}$) within the full correlation space $\mathcal{P}$.

• In the non-signaling regime, the distance is determined by the facets of $\mathcal{P}_{Loc}$ (Theorem 1).
• In the signaling regime, the distance is determined by the "closest" boundary of the target polytope relative to the boundary of the full space $\mathcal{P}$. This requires the additional 112 (for locality) or 112 (for non-signaling) vectors to account for the "corners" and edges of the high-dimensional geometry.

Numerical Analysis

Simulations on random behaviors show that:

• Signaling signatures (f-vectors) dominate the determination of the fractions for most random behaviors (~76-78%).
• Bell-CHSH expressions (e-vectors) are the least likely to be the determining factor for locality in the presence of signaling (only ~0.04% of cases).
• The g, t, and s/z vectors play a significant role (~17-3%), highlighting that simple Bell inequalities are insufficient for a complete analysis when signaling is present.

5. Significance

• Experimental Robustness: The framework allows physicists to analyze real-world Bell experiments (which always have some noise/signaling) without discarding data or making artificial "non-signaling" corrections. It quantifies the "cost" of locality violation.
• Beyond Physics: The method is applicable to causal discovery in fields like economics, psychology, and epidemiology, where signaling (causal influence) is often intrinsic and cannot be ruled out.
• Resource Theory: It reframes non-locality and signaling as resources. The fractional measure quantifies the minimal amount of these resources required to explain observed data.
• Theoretical Advancement: It resolves the geometric complexity of the Bell scenario by providing a complete set of necessary and sufficient linear constraints (the 128/120 vectors) to characterize the local and non-signaling fractions in the general case.

In summary, this paper provides a rigorous, computationally efficient, and universally applicable tool to disentangle genuine quantum non-locality from artifacts caused by signaling, fundamentally extending the scope of Bell's theorem.