Quantum uniformity norms are pullbacks of matrix-valued uniformity norms
2026-06-15 • Information Theory
Information Theory
AI summaryⓘ
The authors show that the quantum uniformity norms recently defined by Bu, Gu, and Jaffe can be understood as versions of known matrix-valued norms introduced by Gowers and Hatami, through a special mathematical embedding. This connection allows them to prove important properties like the triangle inequality and Gowers-Cauchy-Schwarz inequality for these quantum norms, answering an open question by Bu, Gu, and Jaffe. They also relate extreme cases of these norms to structures called Clifford levels, linked to quantum computation concepts developed by Gottesman and Chuang.
quantum uniformity normsWeyl orbit embeddingGowers uniformity normsmatrix-valued normsGowers-Cauchy-Schwarz inequalitytriangle inequalityClifford levelsLeibman polynomial mapsunitary operatorsfinite vector spaces
Authors
Asgar Jamneshan
Abstract
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.