Noncommutative Bohnenblust--Hille Inequality for qudit systems

Abstract

Previous noncommutative Bohnenblust--Hille (BH) inequalities addressed operator decompositions in the tensor-product space $M_2(\mathbb{C}{)}^{\otimes n}$; \emph{i.e.,} for systems of qubits \cite{HCP22,VZ23}. Here we prove noncommutative BH inequalities for operators decomposed in tensor-product spaces of arbitrary local dimension, \emph{i.e.,} $M_K(\mathbb{C}{)}^{\otimes n}$ for any $K\ge 2$ or on systems of $K$-level qudits. We treat operator decompositions in both the Gell-Mann and Heisenberg--Weyl basis, reducing to the recently-proved commutative hypercube BH \cite{DMP} and cyclic group BH \cite{SVZ} inequalities respectively. As an application we discuss learning qudit quantum observables.

30 pages. An old version appeared in arXiv:2301.01438v2 which is replaced by a different paper. Compared with the old submission, this version simplifies some proofs and extends some of the main results to more general qudit systems