Analysing Bell Experiments through Test Factors: Applications to Randomness and Strength of Nonlocality

This work presents practical tools to analyse Bell experiments---experiments demonstrating correlations that defy classical explanations and proving that nature violates local realism. We begin by showing that in the Bell scenario specified by n parties with each party having a choice of m binary-outcome measurements---the (n,m,2) scenario---projecting weakly-signalling settings-conditional outcome distributions onto the smallest-dimensional affine subspace (containing the no-signalling set) via an L^2-distance-minimising map preserves correlators. This result ensures that Bell inequalities written in terms of correlators remain invariant under such projections, and we provide an efficient construction method for the projection operator that avoids computationally costly steps such as Gauss-Jordan elimination and matrix inversion. This will be a useful tool to obtain a readily computable point estimate of a no-signalling distribution when presented with experimental data weakly violating the no-signalling conditions. An important application of a Bell experiment is generation of randomness that can be certified without making assumptions about the inner workings of the measurement devices. We examine the probability estimation framework, a method for certifying randomness in Bell experiments by estimating outcome probabilities conditioned on measurement settings in the presence of adversarial side information, offering a self-contained proof of its asymptotic optimality and clarifying the structure of optimal adversarial strategies. These results are applied concretely in the (2,2,2) Bell scenario, where we analytically characterise no-signalling-constrained attacks and demonstrate the robustness of probability estimation factors against deviations from expected experimental statistics. We further study possible extensions of the framework to quantum-limited adversaries in the (2,2,2) scenario and no-signalling-constrained adversaries in the more general (n,m,k) scenarios. Finally, we analyse two recent experiments demonstrating local operations and shared randomness-based genuine multipartite nonlocality in a three-party network. Traditional standard-deviation-based approaches to demonstrating the presence of nonlocality in Bell experimental data overlook memory effects. Instead, by introducing a computationally efficient polytope approximation technique for optimising test factors while maintaining statistical validity, we construct an adaptation of the prediction-based ratio protocol and use it to quantify the strength of nonlocality by obtaining valid p-values.