Noncommutative Bohnenblust--Hille Inequality for qudit systems

TL;DR

This paper extends the noncommutative Bohnenblust--Hille inequalities from qubit systems to qudit systems with arbitrary local dimensions, providing new bounds for operator decompositions in higher-dimensional quantum systems.

Contribution

It generalizes noncommutative BH inequalities to tensor-product spaces of any local dimension, including both Gell-Mann and Heisenberg--Weyl bases, and connects to recent commutative inequalities.

Findings

Established noncommutative BH inequalities for qudit systems.

Reduced the problem to known commutative hypercube and cyclic group BH inequalities.

Provided implications for learning quantum observables in higher dimensions.

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.