Each "shot" sends N probes (photons or atoms) through a Mach-Zehnder interferometer to measure an unknown phase φ imprinted in one arm. Every individual probe carries one unit of intrinsic quantum (shot) noise, σ₀ = 1 rad.
With N independent, unentangled probes, the noise on the averaged estimate falls only as 1/√N — the standard quantum limit (SQL). With N probes prepared in an entangled or squeezed state (e.g. a NOON state), the collective phase-sensitivity scales as 1/N — the Heisenberg limit (HL), a quadratic improvement.
σ_classical(N) = σ₀ / √N
σ_entangled(N) = σ₀ / N
gain = σ_classical / σ_entangled = √N
- Probes N — more probes per shot narrows both histograms; the entangled one narrows far faster.
- True phase φ — the hidden quantity being estimated; both modes must locate it from noisy shots alone.
- Mode toggle — selects which probe stream animates through the interferometer (classical probes scatter independently at the detector; entangled probes arrive as a tight correlated cluster).
Real-world relevance: squeezed-light injection into LIGO's interferometer arms and spin-squeezed atomic ensembles in atomic clocks are both live implementations of exactly this trade — trading classical shot noise for quantum correlations to push measurement precision closer to the Heisenberg limit.