Pauli Probability Distributions Encode Full Pure-State Quantum Fisher Metric Exactly

The arXiv preprint demonstrates that Pauli-string statistics retain the full local quantum geometry of pure states. By mapping the probability distribution over all 4^N labeled Pauli strings, the authors derive an exact factor-of-two relation between the classical Fisher information and the QFIM without additional assumptions on the Hamiltonian or state family. This holds for arbitrary smooth parameterizations and directly links many-body response functions to classical statistics.

The result connects to quantum metrology and criticality by showing that pairwise Bell measurements on a state and its complex conjugate physically realize the full QFIM extraction. For mixed states the authors separate Bell statistics from Hilbert-Schmidt coordinates and supply an exact positive decomposition of the information gap, clarifying why the metric equivalence is special to pure states.

Prior work on stabilizer Rényi entropies and nonstabilizerness captured only scalar compressions of the same distribution; this paper shows those lose the off-diagonal metric components essential for multiparameter estimation. The labeled Pauli spectrum therefore emerges as a unifying statistical object for geometry, magic, and response.

Next steps include experimental sampling of the distribution on near-term devices to benchmark metrological sensitivity against theoretical bounds and testing whether partial Pauli shadows can approximate the metric with controlled error.