The Pauli Probability Spectrum Carries the Pure-State Quantum Fisher Metric

Abstract

How a quantum state responds to small parameter changes is central to quantum criticality, state distinguishability, and metrological sensitivity. Pauli-string statistics provide a natural many-body representation of quantum states, but it is not evident how much of their local quantum geometry is retained by the corresponding classical probability distribution. We show that, for any smooth family of pure $N$-qubit states, the classical Fisher matrix of the complete labeled Pauli distribution is exactly twice the quantum Fisher information matrix (QFIM). The Pauli spectrum is therefore locally metric-complete, while scalar Rényi functionals retain only compressed information about its parameter-dependent motion. The same distribution has a direct physical realization: pairwise Bell measurements between corresponding sites of a state and its complex conjugate sample the Pauli spectrum and attain the full QFIM of the conjugate-pair state with a fixed transversal readout. For generic mixed states, physical Bell statistics separate from Hilbert--Schmidt-normalized squared Pauli coordinates, and we derive an exact positive decomposition of the resulting Bell-information gap. These results identify the labeled Pauli spectrum as a common statistical structure underlying many-body response, nonstabilizerness, and multiparameter quantum geometry.