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 \(K\geq2\) 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.