Complete set of commuting observables

Summary

A complete set of commuting observables is a family of pairwise commuting observables whose joint eigenvalues determine a state vector up to a phase. Measuring all of them at once fixes the pure state of the system completely. Their common eigenvectors form an orthonormal basis labelled by the tuples of eigenvalues, which is how quantum numbers label states. In operator-algebra terms, the observables generate a maximal abelian von Neumann algebra.