Quantum Cramér-Rao bound on quantum metric as a single-observable uncertainty relation
Phys. Rev. B 114, 235118 – Published 14 October, 2026
DOI: https://doi.org/10.1103/kknq-k13g
Abstract
The Belavkin version of the quantum Cramér-Rao bound dictates that the covariance of any set of operators is bounded by a product of the derivatives of expectation values and the inverse of the quantum metric. We elaborate that because the quantum metric itself is the covariance of the generators of translation in the parameter space, the quantum metric in any dimension is bounded by a product of itself and Berry curvature. The generator formalism further indicates that the bound is equivalent to a single-observable uncertainty relation for systems described by multiple observables, which in the two-observable case recovers the Robertson-Schrödinger uncertainty relation. The momentum space quantum metric and spin operators of three-dimensional (3D) topological insulators under magnetic field are used to demonstrate the validity of these bounds, and their constraints on the spread of Wannier function, absorbance in 2D, and the dielectric function in 3D will be elaborated.
