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

arXiv:2608.21437v1 Announce Type: new Abstract: Stabilizer R\'enyi entropies compress the distribution of squared Pauli expectation values of a pure quantum state into scalar measures of nonstabilizerness. Here we show that, before this compression, the distribution has a remarkably simple information-geometric structure. For any smooth pure-state family of $N$ qubits, its classical Fisher information matrix is exactly twice the quantum Fisher information matrix of the state at every regular point. This information can be accessed directly: by preparing the state together with its complex conjugate and performing pairwise Bell measurements between corresponding qubits, one samples the same Pauli distribution and saturates the quantum Fisher information matrix of the conjugate pair. The readout is fixed and does not depend on the number of estimated parameters or on the particular pure-state sensing model. We show that the origin of this result is more general than Pauli algebra: conjugation identifies the doubled pure state with the vectorized density operator, whose coordinates are real in any Hermitian Hilbert--Schmidt operator basis. The situation changes for mixed states, for which the Bell probabilities no longer coincide with normalized squared Pauli expectation values and the same fixed readout is generally not optimal. We derive an exact positive-semidefinite expression for the missing Fisher information, identifying the components of the state variation that are invisible to Bell sampling. Our results reveal a direct connection between stabilizer R\'enyi statistics, quantum-state geometry, and fixed multiparameter readout.