Local deterministic model of singlet state correlations based on relaxing measurement independence (1007.5518v4)

Abstract: The derivation of Bell inequalities requires an assumption of measurement independence, related to the amount of free will experimenters have in choosing measurement settings. Violation of these inequalities by singlet state correlations, as has been experimentally observed, brings this assumption into question. A simple measure of the degree of measurement independence is defined for correlation models, and it is shown that all spin correlations of a singlet state can be modeled via giving up a fraction of just 14% of measurement independence. The underlying model is deterministic and no-signalling. It may thus be favourably compared with other underlying models of the singlet state, which require maximum indeterminism or maximum signalling. A local deterministic model is also given that achieves the maximum possible violation of the well known Bell-CHSH inequality, at a cost of only 1/3 of measurement independence.

• The paper presents a local deterministic model showing that a mere 14% relaxation in measurement independence reproduces singlet state correlations.
• It employs modified Bell frameworks with deterministic functions and unit vectors to maintain no-signalling while violating Bell inequalities.
• The study contrasts prior models by demonstrating that minimal compromise in experimental free will can achieve maximal Bell-CHSH violations.

Overview of the Local Deterministic Model of Singlet State Correlations

The paper by Michael J.W. Hall presents a significant exploration of the foundational assumptions underlying quantum mechanics and their implications, particularly concerning the violation of Bell inequalities. The focus is on the reinterpretation of measurement independence within the framework of local deterministic models to explain singlet state correlations, traditionally understood through quantum mechanics.

The derivation of Bell inequalities assumes measurement independence, which implies that experimenters can freely choose measurement settings. This paper posits that singlet state correlations can be modeled accurately by relinquishing only a small fraction (14%) of this independence. The model is deterministic and adheres to the no-signalling constraint, distinguishing itself from other models necessitating significant indeterminism or signalling. Furthermore, it demonstrates that a local deterministic model can achieve the maximal violation of the Bell-CHSH inequality with a reduction to two-thirds of measurement independence.

Hall's approach is grounded in a nuanced theoretical examination of measurement independence. This fundamentally involves evaluating how much of this independence must be sacrificed to achieve a deterministic no-signalling model reflecting the experimental realities of singlet state correlations. By introducing a quantifiable metric for the degree of measurement independence, Hall establishes a permissible threshold of relaxation—specifically, allowing for a fraction of 86% measurement independence.

The paper contrasts with other theoretical models like those of Branciard et al. or Toner and Bacon, which either forego determinism or allow nonlocal signalling, respectively. Hall’s work reveals that relaxing the measurement independence assumption is a comparatively less severe compromise, necessitating only a 14% reduction. This result places the model at a balance between theoretical rigor and alignment with empirical quantum mechanical observations.

Hall employs a modified variant of Bell's original model involving unit vectors and deterministic functions to define measurement outcomes. The quantification of measurement independence is explicitly formulated in terms of maximum distances between probability distributions over the model's underlying variable. This explicit characterization serves as a foundation for comparing different degrees of model compromise. Remarkably, the proposed model retains a high degree of experimental free will, denoting a merely fractional dependency between measurement settings and underlying variables.

The paper also extends its scope to investigate Bell inequality violations, emphasizing the degree to which indeterminism must be foregone. It delineates a deterministic no-signalling model capable of maximal violation of the Bell-CHSH inequality by sacrificing only a third of measurement independence. For values of measurement independence below 2/3, specific deterministic models are constructed to demonstrate varying degrees of Bell inequality violation. These models solidify the observation that measurement independence can be relaxed to model quantum correlations without entirely relinquishing fundamental deterministic characteristics.

Hall concludes by equating measurement independence as a manageable constraint, given its minor compromise relative to determinism and signalling in producing accurate models of singlet states. This conclusion opens potential avenues for reevaluating quantum models in terms of relaxed assumptions, thus contributing to the broader discourse on reconciling quantum mechanics with deterministic paradigms.

Overall, Hall's research suggests practical and theoretical implications for future model-building efforts in quantum mechanics, highlighting avenues for exploring new concepts of causality and strategic relaxation of foundational assumptions without sacrificing empirical fidelity.