Phase uncertainty Δφ vs number of atoms N. Climb the ladder; then turn on noise and watch which resource actually wins.
The question: the Heisenberg limit (1/N) beats the standard quantum limit (1/√N) — so should you always reach for the most entangled state? Set the decoherence, pick your N, and read each strategy off the dashed line.
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At zero noise the GHZ line (1/N) is best at every N. Turn up decoherence and it grows a bathtub — a sweet spot, then it climbs. Why does piling more atoms into one entangled state eventually make the measurement worse, and why can plain atoms overtake it?
A GHZ state of N atoms is N times more sensitive to phase — that is the 1/N gain — but it also dephases N times faster: one stray photon scrambles the whole entangled register. So as N grows, the usable coherence time shrinks, and past a noise-set sweet spot N* the fragility outweighs the gain. Independent atoms (SQL) carry no such penalty: each fails on its own, so their 1/√N keeps improving and eventually overtakes the saturated GHZ. Spin-squeezing sits in between — a smaller, more robust gain. The fix for the fragility is not "more entanglement" but protecting it: that is level 4, quantum error correction (toggle +QEC to push N* back out).
SQL: independent probes give Δφ ∝ N−1/2 (this is the same 1/√N wall as the white-noise floor in the anchor-2 explorer). Squeezing lowers the prefactor: ξ/√N, ξ<1 (Caves 1981). A GHZ/NOON state reaches the Heisenberg limit Δφ ∝ N−1 (Giovannetti et al. 2011). Schematic note: the realistic curve here is a teaching cartoon — Δφ ≈ max(1/N, N/N*²), a bathtub with its minimum 1/N* at N = N*. Its asymptotic message is the real result: under uncorrelated dephasing the Heisenberg advantage is lost and one reverts toward 1/√N (Huelga et al. 1997; Demkowicz-Dobrzański et al. 2012). +QEC pushes N* up by protecting coherence.