Shot noise vs. Heisenberg limit for phase estimation
The Cramér–Rao bound limits estimation precision: Var(φ̂) ≥ 1/F where F is the Fisher information. Classical (separable) states achieve the shot-noise limit F = N, giving Δφ ∼ 1/√N. Entangled NOON states achieve the Heisenberg limit F = N², giving Δφ ∼ 1/N — a quadratic improvement.