Quantum Metrology: Heisenberg Limit vs Standard Quantum Limit

Probe EnsembleSQL

Number of probes N 6

True phase φ (unknown) 0.30π

Measurement rate 12 shots/s

Live Estimate

Shots taken

Est. φ

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Estimation error |φ_est − φ_true|

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Theoretical uncertainty Δφ

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Precision gain over SQL

1×

How it works

A quantum-metrology sensor estimates an unknown phase φ (e.g. a magnetic field, a frequency shift, gravity) imprinted on N two-level atomic probes during a Ramsey interrogation, then reads it out with a projective measurement whose outcome probability oscillates with φ.

Standard Quantum Limit (SQL) — N atoms prepared and measured independently (an uncorrelated, "classical" ensemble). Each atom clicks 0/1 with probability p = (1+cosφ)/2. Averaging m repeated shots over N atoms:

p̂ = (total clicks) / (N · m)
φ̂ = arccos(2p̂ − 1)
Δφ_SQL = 1 / √(N·m)   (shot-noise limit)

Heisenberg Limit — the N atoms are prepared in a maximally entangled GHZ state (|000…0⟩ + |111…1⟩)/√2, sometimes called a NOON state in the optical case. A collective parity measurement now oscillates N times faster in φ:

p = (1 + cos(N·φ)) / 2
φ̂ = arccos(2p̂ − 1) / N
Δφ_Heis = 1 / (N·√m)

Because the GHZ fringe is N× narrower, entanglement buys a precision gain of exactly N over the same number of atoms and shots — the basis of atomic clocks, gravimeters and gravitational-wave detectors pushing past classical shot noise (Giovannetti, Lloyd & Maccone, Nat. Photonics 2011). The slider range for φ is capped at π/N so the faster GHZ fringe stays unambiguous (2π/N periodicity).

N — number of probe atoms/photons in the ensemble.
φ — the true phase being sensed; the sim never shows this to the "detector", only the measurement statistics do.
Rate — how many measurement shots are simulated per second.
Mode — toggles between independent atoms (SQL, ring of arrows) and an entangled GHZ ensemble (single fast collective arrow).

Probe / collective spin Outcome "0" Outcome "1"

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