Local hidden-variable theory

​1​(∣01⟩−∣10⟩), where $|0\rangle$∣0⟩ and $|1\rangle$∣1⟩ denote opposite spin projections along a reference axis. In quantum theory, when Alice measures spin along direction $\mathbf{a}$a and Bob along $\mathbf{b}$b, the expectation value of the product of their outcomes is $E(\mathbf{a}, \mathbf{b}) = -\mathbf{a} \cdot \mathbf{b} = -\cos\theta$E(a,b)=−a⋅b=−cosθ, with $\theta$θ the angle between $\mathbf{a}$a and $\mathbf{b}$b. This yields perfect anticorrelation ($E = -1$E=−1) when $\theta = 0$θ=0 and no correlation ($E = 0$E=0) when $\theta = 90^\circ$θ=90∘.^[6] Local hidden-variable models attempt to reproduce these correlations by assigning predetermined outcomes $A(\mathbf{a}, \lambda) = \pm 1$A(a,λ)=±1 for Alice and $B(\mathbf{b}, \lambda) = \pm 1$B(b,λ)=±1 for Bob, determined by a shared hidden variable $\lambda$λ distributed according to $\rho(\lambda)$ρ(λ). The correlation is then $E(\mathbf{a}, \mathbf{b}) = \int A(\mathbf{a}, \lambda) B(\mathbf{b}, \lambda) \rho(\lambda) \, d\lambda$E(a,b)=∫A(a,λ)B(b,λ)ρ(λ)dλ. To match the singlet's perfect anticorrelation at $\theta = 0$θ=0, the model requires $A(\mathbf{a}, \lambda) = -B(\mathbf{a}, \lambda)$A(a,λ)=−B(a,λ) for almost all $\lambda$λ. However, this deterministic relation fails to reproduce the full angular dependence $-\cos\theta$−cosθ across all pairs of directions without violating locality or introducing inconsistencies in outcome assignments.^[6] Under the standard assumption of measurement independence (where $\rho(\lambda)$ρ(λ) is independent of measurement settings), no local deterministic model can reproduce the quantum correlations for the singlet state. However, by relaxing measurement independence—allowing the distribution $\rho(\lambda | \mathbf{x}, \mathbf{y})$ρ(λ∣x,y) to depend on the chosen settings $\mathbf{x}$x and $\mathbf{y}$y—a local deterministic model can exactly match the quantum predictions. An example is Hall's 2010 model, where the hidden variable $\lambda$λ is uniformly distributed on the unit sphere $S^2$S2, with deterministic responses $A(\mathbf{x}, \lambda) = \sgn(\mathbf{x} \cdot \lambda)$A(x,λ)=\sgn(x⋅λ) and $B(\mathbf{y}, \lambda) = -\sgn(\mathbf{y} \cdot \lambda)$B(y,λ)=−\sgn(y⋅λ). The conditional density is $\rho_{\mathbf{x}\mathbf{y}}(\lambda) = \frac{1 + \mathbf{x}\cdot\mathbf{y}}{8(\pi - \theta)}$ρxy​(λ)=8(π−θ)1+x⋅y​ if $\sgn(\mathbf{x}\cdot\lambda) = \sgn(\mathbf{y}\cdot\lambda)$\sgn(x⋅λ)=\sgn(y⋅λ), and $\rho_{\mathbf{x}\mathbf{y}}(\lambda) = \frac{1 - \mathbf{x}\cdot\mathbf{y}}{8\theta}$ρxy​(λ)=8θ1−x⋅y​ otherwise, where $\theta$θ is the angle between $\mathbf{x}$x and $\mathbf{y}$y. This yields the joint probability $p_{\mathbf{x}\mathbf{y}}(a,b) = \frac{1}{4} (1 - ab \, \mathbf{x}\cdot\mathbf{y})$pxy​(a,b)=41​(1−abx⋅y), exactly reproducing the quantum singlet correlations.^[20] This failure under standard assumptions manifests quantitatively through violations of Bell inequalities, such as the Clauser-Horne-Shimony-Holt (CHSH) inequality derived for such bipartite scenarios. The CHSH expression is $|\langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle| \leq 2$∣⟨AB⟩+⟨AB′⟩+⟨A′B⟩−⟨A′B′⟩∣≤2 under local hidden variables, where $A, A'$A,A′ are outcomes for Alice's two settings and $B, B'$B,B′ for Bob's. For the singlet state with measurement settings where Alice chooses directions at $0^\circ$0∘ and $45^\circ$45∘, and Bob at $22.5^\circ$22.5∘ and $67.5^\circ$67.5∘, quantum mechanics predicts a value of $2\sqrt{2} \approx 2.828 > 2$22

​≈2.828>2, demonstrating that no local model can match all correlations simultaneously. Local hidden-variable models inherently preserve the no-signaling condition, as marginal probabilities for one party, such as $P(A = +1 | \mathbf{a}) = \int A(\mathbf{a}, \lambda) \rho(\lambda) \, d\lambda / 2$P(A=+1∣a)=∫A(a,λ)ρ(λ)dλ/2, remain independent of the distant party's choice of setting. Quantum entangled states also satisfy no-signaling, but the joint probability distributions $P(A, B | \mathbf{a}, \mathbf{b})$P(A,B∣a,b) in the singlet state cannot be replicated by local models, as the latter are limited to factorized forms over $\lambda$λ that bound correlations below quantum levels. An additional incompatibility arises from the no-cloning theorem, which states that an unknown quantum state cannot be perfectly copied. In the context of entanglement, a local hidden-variable model would effectively "clone" the predictive information encoded in $\lambda$λ, allowing both parties to deterministically access outcomes without disturbing the shared state, in contradiction to the non-clonable nature of entangled states like the singlet. This argument underscores why local realism cannot accommodate quantum entanglement's monogamy and uniqueness properties.

Limitations and Extensions

No-Go Results