Norm properties of quantum uniformity measures

Abstract:We prove that for $k=4$ the quantum uniformity measures are norms, whereas for every $k\geq5$ the triangle inequality fails. This gives a complete answer to an open question of Bu, Gu, and Jaffe. The $k=4$ case is proved using the Cauchy--Schwarz inequality for the quantum uniformity pairing $\langle\cdot,\cdot\rangle_{Q^4}$ introduced here, together with the triangle inequality in a suitable finite-dimensional Bochner $L^8$ space. For $k\geq5$, we construct systems of linear equations over a finite field and derive a contradiction from the structure theory of norming linear systems.

Mathematical Physics (math-ph); Functional Analysis (math.FA); Quantum Physics (quant-ph)

arXiv:2610.17454 [math-ph]

(or arXiv:2610.17454v1 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2610.17454 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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