Quantum Fisher Information — Cramér-Rao Bound

Probe particles N

Squeezing parameter ξ²

1.00

True phase φ

0.50

Measurements M

100

Quantum Fisher Info F_Q

—

QCR Bound Δφ ≥

—

SQL limit (1/√N)

—

Heisenberg limit (1/N)

—

Quantum Cramér-Rao Bound: Δφ ≥ 1/√(M · F_Q)
For N uncorrelated probes in a coherent state: F_Q = N → Standard Quantum Limit (SQL) Δφ ~ 1/√N. With entangled (GHZ/squeezed) states: F_Q = N² → Heisenberg limit Δφ ~ 1/N. Spin-squeezing reduces variance in one quadrature (ξ² < 1) below shot noise, beating SQL without full entanglement. The Symmetric Logarithmic Derivative (SLD) operator achieves the bound.