Bell States & Entanglement Correlations

The four Bell states are maximally entangled two-qubit states. For each state, the measurement correlation P(same outcome) vs the angle θ between detector axes follows a characteristic curve — violating Bell inequalities (CHSH), which classical correlations cannot. Rotate measurement bases to explore.

|Φ⁺⟩ = (|00⟩ + |11⟩)/√2
P(same) = cos²(θ/2)  |  CHSH: S = 2√2 ≈ 2.828 (violates classical bound S≤2)
Bloch sphere representation (qubit A)
Correlation P(same outcome) vs angle θ
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CHSH parameter S vs noise — red line: classical bound S=2, quantum max S=2√2
Bell / CHSH inequality:
S = |E(a,b) − E(a,b') + E(a',b) + E(a',b')| ≤ 2 (classical hidden variables)
Quantum mechanics: S_max = 2√2 ≈ 2.828 (Tsirelson's bound)

Measurement correlation: for |Φ⁺⟩: E(θ) = cos θ, so P(same) = cos²(θ/2)
For |Φ⁻⟩: E(θ) = −cos θ, |Ψ±⟩: E(θ) = ±cos θ

The key insight: quantum correlations are stronger than any local hidden variable model can produce, as demonstrated by Aspect et al. (1982) and closed-loop tests (Hensen et al. 2015).