Heisenberg limit: δφ ≥ 1/N vs SQL δφ ≥ 1/√N — ratio √N enhancement

For N-probe interferometric phase estimation the Cramér-Rao bound from the quantum Fisher information F_Q gives δφ ≥ 1/√F_Q. Uncorrelated (classical) N-atom states saturate at F_Q = N → δφ ≥ 1/√N (standard-quantum / shot-noise limit). …