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.

翻译：我们证明了Bu、Gu和Jaffe近期引入的量子一致性范数，在Weyl轨道嵌入下，是Gowers与Hatami矩阵值一致性范数的回拉。该恒等关系导出了量子一致性范数的Gowers-Cauchy-Schwarz不等式与三角不等式，从而回答了Bu、Gu和Jaffe提出的一个问题。在极值情形下，该结果用有限向量空间上某些酉值Leibman多项式映射描述了Gottesman与Chuang的Clifford层级。