Quantum Cramér-Rao bound on quantum metric as a single-observable uncertainty relation

Wei Chen

A version of 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 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-Schrödinger 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.

E. R. Caianiello, Wawrzyniec Guz

Hiroshi Nagaoka

Wei Chen

Mathematica files for producing the paper "Quantum Cramér-Rao bound on quantum metric as asingle-observable uncertainty relation".

Changhao Li, Mo Chen, Paola Cappellaro

The power of quantum sensing rests on its ultimate precision limit, quantified by the quantum Cramer-Rao bound (QCRB), which can surpass classical bounds. In multi-parameter estimation, the QCRB is not always saturated as the quantum nature of associated observables may lead to their incompatibility. Here we explore the precision limits of multi-parameter estimation through the lens of quantum geometry, enabling us to experimentally evaluate the QCRB via quantum geometry measurements. Focusing on two- and three-parameter estimation, we elucidate how fundamental quantum uncertainty principles prevent the saturation of the bound. By linking a metric of "quantumness" to the system geometric properties, we investigate and experimentally extract the attainable QCRB for three-parameter estimations.

Seth Lloyd, Vittorio Giovannetti, Lorenzo Maccone