Bell and EPR experiments with signalling data

TL;DR

This paper addresses apparent signalling in Bell and EPR experiments caused by experimental imperfections. It proposes bounded-signalling extensions of LHV and LHS models (SLHV and SLHS_\gamma) and develops both linear-programming and semidefinite-programming tools to test data against these models. It provides analytic corrections to Bell and steering inequalities and constructs optimal witnesses within the bounded-signalling framework, applying them to IBM quantum hardware data and to post-selected detectors. The approach yields a practical means to quantify signalling, inflate classical sets accordingly, and assess the robustness of observed nonlocality or steering under experimental imperfections, while highlighting the foundational importance of no-signalling for device-independent claims.

Abstract

The no-signalling principle is a fundamental assumption in Bell-inequality and quantum-steering experiments. Nonetheless, experimental imperfections can lead to apparent violations beyond those expected from finite-sample statistics. Here, we propose extensions of local hidden variable and local hidden state theories that allow for bounded, operationally quantifiable, amounts of signalling. We show how non-classicality tests can be developed for these models, both through exact methods based on the full set of observed statistics and through corrections to the standard Bell and steering inequalities. We demonstrate the applicability of these methods via two scenarios that feature apparent signalling: an IBM quantum processor and post-selected data from inefficient detectors.

Figures (5)

• Figure 1: Geometric illustration of correlations. (a) Bell scenario. The local polytope (LHV) and the quantum set (Q) are subsets of no-signalling probability distributions. When bounded signalling is permitted, the LHV polytope is expanded into the SLHV polytope. (b) EPR scenario. The unsteerable assemblages (LHS) are a subset of no-signalling assemblages. Bounded signalling inflates the former into the SLHS$_\gamma$ set. Both the SLHV and SLHS$_\gamma$ sets extend both within and outside the no-signalling subspace.
• Figure 2: Standard quantum CHSH test but when post-selecting on successful detections with outcome efficiencies $\eta_{0}$ and $\eta_{1}$. The heat map represents how well the statistics can be approximated by an SLHV model, quantified through the visibility parameter $v$.
• Figure 3: Critical visibility for witnessing high-dimensional steering. The solid lines represent the bounds for certifying entanglement, and the dashed lines represent the bounds for certifying Schmidt number $3$. Here, the blue and red represent the bounds obtained using the non-adjusted witness and the adjusted witness, respectively.
• Figure 4: Signalling parameter $\alpha$ plotted as a function of efficiencies $\eta_{0}$ and $\eta_{1}$ as described in \ref{['eq:Alpha_eta_function']}.
• Figure 5: The CHSH parameter plotted as a function of detection efficinceis $\eta_{0}$ and $\eta_{1}$ of the post-selection of the behaviour $p(a,b|x,y)=\frac{1}{4}\left(1+(-1)^{a+b+xy}\right)$. The heat map represents the visibility of the post-selectd behaviour in the SLHV model with $\alpha^{ax}_{yy'}=|p(a|x,y)-p(a|x,y')|$ and $\beta^{by}_{xx'}=|p(b|x,y)-p(b|x',y)|$.