A Family of Local Deterministic Models for Singlet Quantum State Correlations

This work investigates the implications of relaxing the measurement independence assumption in Bell’s theorem by introducing a new class of local deterministic models that account for both particle preparation and measurement settings. Our model reproduces the quantum mechanical predictions under the assumption of relaxed measurement independence, demonstrating that the statistical independence of measurement settings does not necessarily preclude underlying correlations. Our findings highlight the nuanced relationship between local determinism and quantum mechanics, offering new insights into the nature of quantum correlations and hidden variables.

💡 Research Summary

The paper investigates the consequences of relaxing the measurement‑independence assumption that underlies Bell’s theorem, and it introduces a new family of local deterministic hidden‑variable models capable of reproducing the singlet‑state correlations predicted by quantum mechanics. The authors begin by recalling the standard Bell‑CHSH scenario: a source emits pairs of spin‑½ particles in the singlet state, Alice and Bob choose measurement directions x and y, and the quantum expectation value is ⟨AB⟩ = −cos ϕₓᵧ, where ϕₓᵧ is the angle between the two directions. In the traditional Bell framework, hidden variables λ are assumed to influence only the preparation of the particle pair, while the measurement settings are taken to be statistically independent of λ. This independence is a crucial ingredient in deriving Bell inequalities such as |E(x,y)−E(x,z)| ≤ 1+E(y,z).

The authors argue that this assumption need not hold. They introduce an enlarged hidden‑variable set Λ = (λ, α, β), where λ still describes the intrinsic state of the particle pair, but α and β are additional hidden variables that determine the actual measurement settings chosen by Alice and Bob. Consequently, the expectation value becomes

E(x,y) = ∫ dΛ ρ(Λ) w(Λ;x,y) A(x,Λ) B(y,Λ) / ∫ dΛ ρ(Λ) w(Λ;x,y),

where w(Λ;x,y) is a weight function that selects only those hidden‑variable configurations compatible with the chosen settings. By defining a normalized distribution ρ̃(Λ) = ρ(Λ) w(Λ;x,y)/∫ρ(Λ) w(Λ;x,y)dΛ, the expression simplifies to the familiar form E(x,y)=∫ρ̃(Λ)A(x,Λ)B(y,Λ)dΛ.

The central construction of the paper is a one‑parameter family of models indexed by a real parameter γ. The authors propose a specific functional form for the joint density

ρ̃(λ,α,β) = f(α·λ, β·λ) |α·λ| |β·λ| δ(α−x) δ(β−y),

where f is a non‑negative homogeneous function of degree k = 2(γ−1). The measurement functions are taken to be deterministic sign functions: A(x,λ)=sgn(x·λ) and B(y,λ)=−sgn(y·λ). After integrating over α and β, the expectation value reduces to

E(x,y)=−∫ dλ f(x·λ, y·λ) |x·λ| |y·λ|.

Normalization imposes

∫ dλ f(x·λ, y·λ) |x·λ| |y·λ| = 1.

Choosing a coordinate system with x = (1,0,0) and y = (cos ϕₓᵧ, sin ϕₓᵧ, 0), the authors evaluate the integrals analytically for the case where f has the piecewise form

f(u,v)=c₁|uv|(u²+v²)^γ if sgn(u)=sgn(v),
f(u,v)=c₂|uv|(u²+v²)^γ if sgn(u)≠sgn(v).

The constants c₁ and c₂ are fixed by the normalization condition and by demanding that the model reproduce the quantum prediction E(x,y)=−cos ϕₓᵧ. The degree of homogeneity k must satisfy k > −4, which translates into γ > −1. For generic γ the integrals must be evaluated numerically; however, for γ = 0 (k = −2) the integrals are elementary, yielding

c₁(ϕ)= (1+cos ϕ) /