Norm properties of quantum uniformity measures

We prove that for $k=4$ the quantum uniformity measures are norms, whereas for every $k\ge 5$ the triangle inequality fails. This gives a complete answer to an open question of Bu, Gu, and Jaffe. The $k=4$ case is proved using the Cauchy–Schwarz inequality for the quantum uniformity pairing $⟨\cdot ,\cdot {⟩}_{Q^4}$ introduced here, together with the triangle inequality in a suitable finite-dimensional Bochner $L^8$ space. For $k\ge 5$, we construct systems of linear equations over a finite field and derive a contradiction from the structure theory of norming linear systems.