Sharp uncertainty relations for number and angle

TL;DR

This paper derives sharp uncertainty relations for conjugate variables like number and angle, showing how measurement and preparation uncertainties coincide and depend on chosen metrics, with both numerical and analytical bounds.

Contribution

It introduces a new approach to quantify uncertainty bounds for number and angle variables, considering different metrics and providing explicit lower bounds.

Findings

Uncertainty bounds depend only on metric choices.

Measurement and preparation uncertainties coincide for these variables.

New method for certified lower bounds in ground state problems.

Abstract

We study uncertainty relations for pairs of conjugate variables like number and angle, of which one takes integer values and the other takes values on the unit circle. The translation symmetry of the problem in either variable implies that measurement uncertainty and preparation uncertainty coincide quantitatively, and the bounds depend only on the choice of two metrics used to quantify the difference of number and angle outputs, respectively. For each type of observable we discuss two natural choices of metric, and discuss the resulting optimal bounds with both numerical and analytic methods. We also develop some simple and explicit (albeit not sharp) lower bounds, using an apparently new method for obtaining certified lower bounds to ground state problems.