  Uncertainty relations are a fundamental feature of quantum mechanics. How can
these relations be found systematically? Here we develop a semidefinite
programming hierarchy for additive uncertainty relations in the variances of
non-commuting observables. Our hierarchy is built on the state polynomial
optimization framework, also known as scalar extension. The hierarchy is
complete, in the sense that it converges to tight uncertainty relations. We
improve upon upper bounds for all 1292 additive uncertainty relations on up to
nine operators for which a tight bound is not known. The bounds are
dimension-free and depend entirely on the algebraic relations among the
operators. The techniques apply to a range of scenarios, including Pauli,
Heisenberg-Weyl, and fermionic operators, and generalize to higher order
moments and multiplicative uncertainty relations.
