Quantum metrology - 2D estimation lab

Phase estimation with a Mach-Zehnder interferometer

Each shot sends N probes (photons or atoms) through the interferometer and reads the output ports. Working at the steepest point of the fringe, the unknown offset δ changes the probability of the '+' port. With independent probes every probe gives P(+) = (1 − sinδ)/2, one Bernoulli trial with Fisher information 1, so F = η N per shot and the error obeys the standard quantum limit σ = 1/√(η N M) for M shots.

A NOON state (all N probes in a superposition of both arms) accumulates the phase N times faster: P(+) = (1 − V sin(Nδ))/2 with visibility V = η^N. The Fisher information per shot is F = N² V² cos²(Nδ) / (1 − V² sin²(Nδ)), giving the Heisenberg limit σ = 1/(N√M) without loss. Loss destroys it exponentially: F peaks near N* = −1/lnη (graph 2). Squeezed light keeps the 1/√N scaling but lowers the noise variance by e^−2r; loss mixes vacuum noise back in: σ² = (η e−2r + 1 − η) / (η N M).

Estimates here come from the maximum-likelihood inversion of the exact binomial outcome statistics, repeated many times to measure σ directly; the three in-page checks compare these measured widths with the Cramer-Rao bounds.

Honest simplifications

NOON states are assumed perfect (a loss of any one probe wipes the signal), the phase is known to lie inside the window of width π/N (a prior from a coarser measurement), and the offset is assumed biased to the steepest fringe point. The squeezed-light estimator is modelled as Gaussian rather than from photon counting. Real implementations are LIGO squeezed vacuum and spin-squeezed atomic clocks.

• Lossless preset: the NOON histogram is sqrt(N) = 4 times narrower than the classical one.
• Switch to 5 % and 20 % loss and watch the NOON advantage collapse while squeezing keeps helping.
• Sweep N in graph 1: classical and NOON lines have slopes -1/2 and -1 until loss bends the NOON curve.