Norm properties of quantum uniformity measures
Abstract
We give a complete classification of the norm properties of quantum uniformity measures, introduced by Bu, Gu, and Jaffe to analytically characterize the Clifford hierarchy. Resolving an open question they posed, we show that quantum uniformity measures define norms precisely for k=2,3,4. In particular, we establish the triangle inequality for k=4, whereas for every k≥5 it fails in some dimension. The higher-order result is proved using the structure theory of norming systems of linear equations over finite fields. We also study the geometry of quantum uniformity measures, establishing explicit bounds on the modulus of convexity whenever they define norms. Finally, we give a complete classification of the norm properties of matrix-valued Gowers uniformity functionals, showing that only k=2 defines a norm.
Related
- Some properties of Fourier quasicrystals and measures on a strip
- Time-optimal quantum dynamics under fixed-norm generators
- Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity
- Dirac measures can be quantum limits on the Sierpinski gasket
- Half-delocalization for semiclassical measures of split higher-dimensional quantum cat maps
BibTeX
@misc{unal2026norm,
title = {{Norm properties of quantum uniformity measures}},
author = {Berke Ünal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17454},
eprint = {2610.17454},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17454},
}
RIS
TY - UNPB
TI - Norm properties of quantum uniformity measures
AU - Berke Ünal
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.17454
UR - https://arxiv.org/abs/2610.17454
AB - We give a complete classification of the norm properties of quantum uniformity measures, introduced by Bu, Gu, and Jaffe to analytically characterize the Clifford hierarchy. Resolving an open question they posed, we show that quantum uniformity measures define norms precisely for k=2,3,4. In particular, we establish the triangle inequality for k=4, whereas for every k≥5 it fails in some dimension. The higher-order result is proved using the structure theory of norming systems of linear equations over finite fields. We also study the geometry of quantum uniformity measures, establishing explicit bounds on the modulus of convexity whenever they define norms. Finally, we give a complete classification of the norm properties of matrix-valued Gowers uniformity functionals, showing that only k=2 defines a norm.
ER -