Uncertainty relations for general phase spaces

TL;DR

This paper develops a unified framework for deriving uncertainty relations across various quantum systems with different phase space structures, accommodating diverse metrics and deviations.

Contribution

It introduces a general method to obtain uncertainty relations for observables linked by Fourier transform on arbitrary locally compact abelian groups.

Findings

Uncertainty bounds are equal for measurement and preparation in all cases.

A straightforward method for determining optimal uncertainty bounds is provided.

Applicable to standard and quantum information phase spaces.

Abstract

We describe a setup for obtaining uncertainty relations for arbitrary pairs of observables related by Fourier transform. The physical examples discussed here are standard position and momentum, number and angle, finite qudit systems, and strings of qubits for quantum information applications. The uncertainty relations allow an arbitrary choice of metric for the distance of outcomes, and the choice of an exponent distinguishing e.g., absolute or root mean square deviations. The emphasis of the article is on developing a unified treatment, in which one observable takes values in an arbitrary locally compact abelian group and the other in the dual group. In all cases the phase space symmetry implies the equality of measurement uncertainty bounds and preparation uncertainty bounds, and there is a straightforward method for determining the optimal bounds.