Quantum Fisher Information & Metrological Bound

Number of probes N 20 Squeezing ξ² 0.50 Phase θ 0.50 Decoherence γ 0.00

SQL: 1/N = —

HL: 1/N² = —

Squeezed: —

GHZ: —

Cramér-Rao Bound

The precision of phase estimation obeys Δθ² ≥ 1/F_Q, where F_Q is the Quantum Fisher Information. For N independent classical probes (coherent state): F_Q = N (SQL). For entangled GHZ state: F_Q = N² (Heisenberg limit). Spin squeezing beats SQL: F_Q = N/ξ².

Decoherence degrades the advantage: GHZ becomes fragile.