Quantum Metrology: Heisenberg Limit vs Standard Quantum Limit

This simulator compares the two fundamental precision limits of quantum-enhanced sensing side by side. In "SQL" mode, N atomic probes are prepared and measured independently, and the best achievable phase precision scales as 1/√(N·shots) — the standard quantum limit that classical shot-noise-limited atomic clocks and interferometers are bound to. Switch to "GHZ" mode and the same N atoms are instead entangled into a single Greenberger–Horne–Zeilinger state; the collective measurement fringe oscillates N times faster in the unknown phase, pushing the achievable precision down to 1/(N·√shots) — the Heisenberg limit. A live Monte Carlo measurement loop draws real binomial-outcome samples each "shot", accumulates a running phase estimate, and reports the empirical error alongside the theoretical uncertainty bound for both regimes, making visible exactly how entanglement buys metrological precision without spending more probes.