Uncertainty relations for characteristic functions

TL;DR

This paper introduces a generalized uncertainty relation for characteristic functions in quantum mechanics, extending Heisenberg's principle, with applications in quantum optics, cosmology, and measurement techniques.

Contribution

It presents a new uncertainty relation for characteristic functions, applicable to various quantum systems and measurement scenarios, surpassing traditional Heisenberg bounds.

Findings

The ChUR is saturated for wavefunctions with periodic Dirac combs.

Applications include constraining quantum optical measurements with arbitrary apertures.

A method for directly measuring characteristic functions using an auxiliary qubit is proposed.

Abstract

We present the uncertainty relation for the characteristic functions (ChUR) of the quantum mechanical position and momentum probability distributions. This inequality is more general than the Heisenberg Uncertainty Relation, and is saturated in two extremal cases for wavefunctions described by periodic Dirac combs. We further discuss a broad spectrum of applications of the ChUR, in particular, we constrain quantum optical measurements involving general detection apertures and provide the uncertainty relation that is relevant for Loop Quantum Cosmology. A method to measure the characteristic function directly using an auxiliary qubit is also briefly discussed.