Phase Multiplexing for Non-Adaptive Dead-Spot Suppression in GHZ Parity Metrology

Phase-multiplexed GHZ metrology is a non-adaptive quantum sensing protocol designed to suppress "dead spots"—regions of zero sensitivity—inherent in standard Greenberger-Horne-Zeilinger (GHZ) states under X-basis parity readout. In a single-branch GHZ interferometer, the Classical Fisher Information (CFI) vanishes wherever the signal derivative is zero, rendering the sensor locally blind at fringe extrema. While traditional mitigation involves real-time feedback like phase-alternation servo locking or adaptive Bayesian estimation, this protocol uses a fixed-hardware approach. The scheme prepares three GHZ probes in parallel with equally spaced fixed phase offsets of 0, 2π/3, and 4π/3. Each branch is measured independently in the parity basis, and their CFI channels are combined with an equal shot budget. This configuration ensures that at any parameter value θ, at least one branch remains far from a fringe extremum, resulting in a total CFI (Ftotal) that is uniform across the entire [0,2π) range to within 2.3% peak-to-peak variation. This method adapts classical phase-shifting principles from the mid-1980s to quantum-enhanced probes. A key operational characteristic is its non-adaptive nature, which eliminates the need for real-time feedback loops or posterior distribution updates. However, this involves a specific engineering trade-off: it triples the hardware overhead for state preparation and readout, which may be more costly than software-based feedback in systems where stable local references are available. Scientifically, the protocol remains consistent with established no-go theorems for noisy quantum metrology. Under uncorrelated Markovian dephasing, the scheme provides only a constant-factor improvement over the Standard Quantum Limit (SQL) and does not achieve Heisenberg scaling. The mean Fisher information is conserved across the parameter space, confirming that the multiplexing redistributes rather than generates information. The performance is validated using resolution-independent metrics, such as the fraction of the parameter range operating below specific sensitivity thresholds, to avoid numerical artefacts associated with sampling true zeros. Keywords: GHZ Metrology, Classical Fisher Information (CFI), Phase Multiplexing, Dead-Spot Suppression, Non-Adaptive Sensing, X-Basis Parity, Standard Quantum Limit (SQL), Quantum Precision Limits.