A weak external phase signal φ(t) — a magnetic field, a gravitational gradient, a frequency offset — is imprinted on N quantum sensor nodes during an interrogation window. Two strategies estimate φ from the same N nodes:
Independent (product) probes: F_prod = N · e^(-2γ)
GHZ-entangled probes: F_GHZ = N² · e^(-2Nγ)
Phase uncertainty (Cramér-Rao): Δφ = 1 / √F
Standard Quantum Limit: Δφ_SQL = 1 / √N (γ→0, unentangled)
Heisenberg limit: Δφ_HL = 1 / N (γ→0, entangled)
With zero decoherence (γ = 0), entangling all N nodes into one GHZ state gives quadratically more Fisher information than N independent probes — the Heisenberg-limit advantage. But a GHZ state dephases N times faster than a single probe, because every node's local noise attacks the shared coherence at once. Push γ up (or pick a lossier topology — a ring or chain has to relay entanglement hop-by-hop, effectively raising the noise each node accumulates) and the exponential penalty N²e^(-2Nγ) collapses below N·e^(-2γ): past a critical γ, plain independent sensors beat the "smarter" entangled network. This crossover is the central result of decoherence-limited quantum metrology (Huelga et al., 1997) and the reason real quantum sensor networks (SQUID arrays, atomic clocks, gravimeter networks) only entangle small clusters rather than the whole array.
- N slider — number of sensor nodes sharing the interrogation.
- γ slider — per-node dephasing rate over the measurement window.
- Topology — star / ring / chain changes how efficiently entanglement is distributed (an effective multiplier on γ for the entangled case).
- GHZ-Entangled toggle — switch between the entangled (GHZ) and independent (product-state) estimation strategy on the same nodes.
- Drag the network view — rotate the ring of sensor nodes around the signal source; the lower panel plots Δφ(γ) for both strategies with your current point marked.