Quantum Metrology — Heisenberg Scaling & Fisher Information
How entanglement beats the standard quantum limit for phase estimation
Quantum metrology uses quantum resources to estimate parameters with precision beyond classical limits. For N independent measurements, classical (shot-noise) precision scales as δφ ~ 1/√N (Standard Quantum Limit). With entanglement (e.g., N00N states |N,0⟩+|0,N⟩ or GHZ states), precision reaches the Heisenberg limit: δφ ~ 1/N. The Quantum Fisher Information F_Q bounds precision via the Cramér-Rao inequality: Var(φ̂) ≥ 1/F_Q. For N00N states, F_Q = N² vs. F_Q = N for separable states — a quadratic improvement. This simulation shows Monte Carlo phase estimation and the precision scaling for each state type.