Uncertainty Relations for Approximation and Estimation

TL;DR

This paper introduces a new, versatile uncertainty relation inequality that unifies approximation and estimation of observables and parameters, connecting weak values, Fisher information, and the Cramér-Rao bound.

Contribution

It presents a generalized uncertainty inequality that encompasses standard relations and links weak values with classical Fisher information for the first time.

Findings

Optimal proxy functions are given by Aharonov's weak value.

The inequality reduces to the Cramér-Rao bound in estimation.

It unifies position-momentum and time-energy uncertainty relations.

Abstract

We present a versatile inequality of uncertainty relations which are useful when one approximates an observable and/or estimates a physical parameter based on the measurement of another observable. It is shown that the optimal choice for proxy functions used for the approximation is given by Aharonov's weak value, which also determines the classical Fisher information in parameter estimation, turning our inequality into the genuine Cram{\'e}r-Rao inequality. Since the standard form of the uncertainty relation arises as a special case of our inequality, and since the parameter estimation is available as well, our inequality can treat both the position-momentum and the time-energy relations in one framework albeit handled differently.