Quantum uniformity norms are pullbacks of matrix-valued uniformity norms

We show that the quantum uniformity norms recently introduced by Bu, Gu, and Jaffe are the pullbacks, under the Weyl orbit embedding, of the matrix-valued uniformity norms of Gowers and Hatami. This identification yields the Gowers-Cauchy-Schwarz inequality and the triangle inequality for the quantum uniformity norms, answering a question of Bu, Gu, and Jaffe. In the extremal regime, it describes the Clifford levels of Gottesman and Chuang in terms of certain unitary-valued Leibman polynomial maps on finite vector spaces.