Cohomological Origin of Unitary-Class Optimality in Quantum Curvature Metrology

We develop a unified geometric framework in which the quantum geometric tensor (QGT) simultaneously governs both the metrology of gauge-invariant curvature and the symmetry classification of random-matrix ensembles. The symmetric Fubini–Study sector, generated by the Jordan product of symmetric logarithmic derivatives (SLDs), determines the quantum Fisher information and therefore the information gain associated with closed-loop curvature acquisition. The antisymmetric Berry sector, classified by the de Rham cohomology class of the SLD connection, determines whether the accumulated holonomy phase is linearly observable. We show that time-reversal symmetry acts as a topological gate on the readable Berry channel: the holonomy survives only in the unitary Wigner–Dyson class (GUE), where the cohomology class is nontrivial. This converts the previously phenomenological time-reversal “gate factor” into a cohomological indicator. In the symplectic class (GSE), the non-Abelian obstruction is identified with the Jordan associator curvature and appears operationally as a Holevo incompatibility penalty. The resulting cost-efficiency functional is shown to be genuinely U-shaped across the Dyson classes, with a strict minimum at GUE arising from the non-monotonic topology of the readable holonomy sector. We further establish a bridge between the QGT and thermodynamic fluctuation geometry, derive curvature-statistics scaling laws from Wigner level repulsion, and analyze the deformation of the Kähler structure under decoherence through the Bures metric and mean Uhlmann curvature. The framework unifies Berry geometry, quantum metrology, random-matrix universality, and information geometry into a single cohomological description of curvature acquisition.