Quantum uniformity norms are pullbacks of matrix-valued uniformity norms

We show that the quantum uniformity norms introduced by Bu, Gu, and Jaffe are the pullbacks of the matrix-valued uniformity norms of Gowers and Hatami under the map $S\mapsto(a\mapsto W_aSW_a^*)$, which we call the Weyl-orbit embedding. This identification yields the Gowers--Cauchy--Schwarz inequali