SU(2) uncertainty limits

TL;DR

This paper investigates uncertainty relations for the SU(2) group, identifying limitations of existing bounds and calculating sharp uncertainty limits for specific quantum states, revealing that pure states can reach maximum uncertainty.

Contribution

It systematically derives sharp uncertainty bounds for SU(2) observables and explores the behavior of pure states in relation to these limits.

Findings

Existing relations are not sharp or saturable.

Calculated variance bounds for spin 1 and spin 3/2 states.

Pure states can attain maximum uncertainty limits.

Abstract

Although progress has been made recently in defining nontrivial uncertainty limits for the SU(2) group, a description of the intermediate states bound by these limits remains lacking. In this paper we enumerate possible uncertainty relations for the SU(2) group that involve all three observables and that are, moreover, invariant under SU(2) transformations. We demonstrate that these relations however, even taken as a group, do not provide sharp, saturable bounds. To find sharp bounds, we systematically calculate the variance of the SU(2) operators for all pure states belonging to the $N=2$ and $N=3$ polarisation excitation manifold (corresponding to spin 1 and spin 3/2). Lastly, and perhaps counter to expectation, we note that even pure states can reach the maximum uncertainty limit.