Eberhard bound: minimum detector efficiency 2(√2−1) for loophole-free Bell violation — E8 Intelligence Research

FINDING: The Eberhard bound η ≥ 2(√2 − 1) ≈ 0.8284 is the minimum detector efficiency for a maximally entangled two-photon state to violate a Bell inequality without the fair-sampling (detection) loophole. | MATH: η_crit = 2(√2 − 1) = 2√2 − 2 ≈ 0.828427. Derivation: for the singlet |ψ⁻⟩ = (|HV⟩ − |VH⟩)/√2, the CHSH correlator is E(θ) = −cos 2θ. With detection efficiency η, the measured correlator is E_meas = η²·E (both photons must be detected). The CHSH parameter S = |E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′)|. Optimizing over angles gives S_max = 2√2·η². Bell violation requires S > 2, so η² > 1/√2, giving η > 2^(−1/4) ≈ 0.8409 for the naive CHSH form. Eberhard's tighter inequality (using the full coincidence-counting statistics, not just correlators) relaxes this to η > 2(√2 − 1) ≈ 0.8284. | CONNECTION: 2(√2 − 1) = 2/φ²·(something)? Check: √2 − 1 = 0.4142 = 1/(1+√2) = tan(π/8). Indeed tan(π/8) = √2 − 1. So η_crit = 2·tan(π/8). The angle π/8 = 22.5° is the optimal CHSH measurement angle. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com