Sum uncertainty relations for compact classical Lie algebras - Namrata Shukla

The sum uncertainty relations are useful because they provide us with a state-independent non-trivial bound. We describe the construction of sum uncertainty relations for compact classical Lie algebras of the type su(n), so(2n), so(2n+1) and sp(n). Then we present resulting uncertainty relations explicitly for the su(2), su(3) and su(4) cases. In order to verify the bounds for general irreps of su(2) and su(3) and su(4), we develop and run a computer program that checks that the relations are correct for general states. Our method uses as a starting point the quadratic Casimir operator of the algebra, which can be recognized as a sum of variances of the generators using elementary observations. We discuss what type of states saturate the su(n) bound, and compare these states with states that saturate more familiar products of uncertainties. Our method is valuable for extending sum uncertainty relations to su(n) and beyond su(n) algebras.