  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.
