Quantum Metrology: When Decoherence Breaks the Heisenberg Limit

Probe & Noise

Particles in GHZ state N 10

Dephasing rate Γ (1/T₂, ms⁻¹) 0.05

Ramsey wait time t (ms) 1.0

Strategy

Live Readouts

Phase precision Δφ (this N, with noise)

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Coherence e^(−NΓt)

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Optimal N* = 1/(Γt)

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How it works

A Ramsey interferometer estimates an unknown phase φ (e.g. a magnetic field or a clock detuning) accumulated during a free-evolution time t. The precision of any unbiased estimator is bounded by the quantum Cramér-Rao bound, Δφ ≥ 1/√F, where F is the Fisher information carried by the probe state.

N independent atoms (classical):     F = N            → Δφ_SQL  = 1/√N
N-atom GHZ state (ideal, no noise):  F = N²           → Δφ_HL   = 1/N
N-atom GHZ state + dephasing:        F = N² e^(−2NΓt) → Δφ(N)   = e^(NΓt) / N

Individual dephasing at rate Γ (loss of single-particle coherence, T₂ = 1/Γ) attacks an N-particle GHZ state N times faster than a single particle, because the whole cat-state decoheres the instant any one of its N particles does. That exponential penalty grows with N while the Heisenberg gain only grows linearly, so Δφ(N) has a minimum, not a monotonic improvement:

d/dN [e^(NΓt)/N] = 0  →  N* = 1/(Γt)

N slider — size of the entangled (or independent) probe ensemble.
Γ, t sliders — set the decoherence budget NΓt accumulated during the measurement.
Entangled / Independent — switches the Fisher information formula used for the live curve and the 3D cluster.
Jump to optimal N* — snaps N to the decoherence-limited sweet spot; past it, more entanglement actively hurts precision.

This is exactly why real atomic clocks and entangled magnetometers (Huelga et al., PRL 1997; Demkowicz-Dobrzański & Maccone, PRL 2014) cannot simply keep growing N for ever-better sensitivity — decoherence sets a hard ceiling on how much of the Heisenberg limit is actually usable, and the true optimum trades entanglement size against T₂.

GHZ / independent probe atoms SQL 1/√N (no entanglement) Heisenberg 1/N (ideal, no noise) With dephasing (current Γ,t)

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