Multiparameter Quantum Estimation and Degeneracy Structure in Three-Flavor Neutrino Oscillations

We investigate three-flavor neutrino oscillations within the framework of quantum estimation theory using the quantum Fisher information matrix (QFIM). We analyze all six independent elements of the QFIM for the parameters θ 23, δ CP, and Δm 2 31, together with the corresponding quantum Cramér--Rao bounds and normalized parameter correlations. To connect the intrinsic quantum sensitivity of the neutrino state with experimentally accessible information, we further compare the QFI with the classical Fisher information (CFI) associated with flavor-probability measurements. We find that the difference between the quantum and classical information bounds is strongly dependent on the oscillation parameter and L/E, demonstrating that the efficiency with which the available quantum information is accessed depends on the measurement. We also investigate probability-degenerate solutions in the (θ 23,δ CP) parameter space. Although different parameter sets can produce identical ν μ toν e appearance probabilities at specific L/E values, their corresponding quantum states can remain distinct, as demonstrated by the quantum fidelity. The QFIM differences further reveal distinct local quantum-information geometries for these probability-degenerate solutions, while the corresponding CFIM differences show a parameter and L/E dependent degree of distinguishability accessible through the flavor-probability measurement. Our results demonstrate that QFI and CFI provide complementary information for neutrino parameter estimation, with the QFIM serving as a measurement-independent quantum benchmark and the CFI characterizing the information accessible through a specified measurement.