Quantum Cram\'{e}r-Rao bound on quantum metric as a multi-observable uncertainty relation

TL;DR

This paper explores a quantum Cramér-Rao bound involving quantum metric and Berry curvature, revealing a multi-observable uncertainty relation that generalizes known uncertainty principles and is demonstrated in topological insulators.

Contribution

It introduces a multi-observable uncertainty relation based on quantum metric and Berry curvature, extending the quantum Cramér-Rao bound to multiple operators.

Findings

Quantum metric is bounded by itself and Berry curvature.

The bound generalizes the Robertson-Schrödinger uncertainty relation.

Application to topological insulators confirms the theoretical bounds.

Abstract

A version of quantum Cram\'{e}r-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 quantum metric. We elaborate that because quantum metric itself is the covariance of the generators of translation in the parameter space, 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 multi-observable uncertainty relation, which in the two-operator case recovers the Robertson-Schr\"{o}dinger uncertainty relation. The momentum space quantum metric and spin operators of three-dimensional topological insulators under magnetic field are used to demonstrate the validity of the three-operator version of these bounds.