Quantum Metrology

Fisher information, Cramér-Rao bound, and the Heisenberg limit

SQL: δφ = 1/√N
Heisenberg: δφ = 1/N
QFI-based: δφ = 1/√F_Q
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The quantum Cramér-Rao bound δφ ≥ 1/√(n·F_Q) limits phase estimation precision. Classical (separable) states reach the SQL δφ = 1/√N. Entangled states can reach the Heisenberg limit δφ = 1/N, a quadratic improvement. The plot shows the Fisher information landscape as a function of phase, with the probability distribution for measurement outcomes. Squeezing redistributes quantum noise to improve sensitivity.