Noncommutative Bohnenblust--Hille inequalities

DOI10.1007/S00208-023-02680-0arXiv2210.14468MaRDI QIDQ6415163

Haonan Zhang, Alexander Volberg

Publication date: 26 October 2022

Abstract: Bohnenblust--Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree cite{defant2019fourier}. Such inequalities have found great applications in learning low degree Boolean functions cite{eskenazis2022learning}. Motivated by learning quantum observables, a quantum counterpart of Bohnenblust--Hille inequality for Boolean cubes was recently conjectured in cite{RWZ22}. In this paper, we prove such noncommutative Bohnenblust--Hille inequalities with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for polynomials of low degrees. Using similar ideas, we also study learning problems of quantum observables of low degree and Bohr's radius phenomenon on quantum Boolean cubes.

Mathematics Subject Classification ID

Norms (inequalities, more than one norm, etc.) of linear operators (47A30) Boolean functions (06E30) Quantum information, communication, networks (quantum-theoretic aspects) (81P45)

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