State-independent uncertainty relations

The standard state-dependent Heisenberg-Robertson uncertainty-relation lower bound fails to capture the
quintessential incompatibility of observables as the bound can be zero for some states. To remedy this problem,
we establish a class of tight (i.e., inequalities are saturated) variance-based sum-uncertainty relations derived
from the Lie algebraic properties of observables and show that our lower bounds depend only on the irreducible
representation assumed carried by the Hilbert space of state of the system. We illustrate our result for the cases
of the Weyl-Heisenberg algebra, special unitary algebras up to rank 4, and any semisimple compact algebra. We
also prove the usefulness of our results by extending a known variance-based entanglement detection criterion.