Intrinsic Bohnenblust--Hille Inequalities for Local Qudit Systems

Abstract

We study dimension-free Bohnenblust--Hille inequalities for local operators on systems of $K$-level qudits. For the operator space of support at most $d$, uniformly over all tensor-product orthonormal operator bases, we prove a Bohnenblust--Hille inequality with the optimal exponent $2d/(d+1)$ and a constant of order $O(\sqrt K)^d$. Our proof is intrinsic, based on a one-site scalarization and block-transversal decoupling, and does not rely on a scalarization to the cyclic group or on a Remez-type argument. We complement the upper bound by showing that the asymptotic exponential Bohnenblust--Hille base satisfies $ cK^{1/4}\le β(K)\le C\sqrt K $ with universal constants $c,C>0$. In this sense, the present work may be viewed as continuing the line of work of Slote--Volberg--Zhang from a complementary intrinsic viewpoint. We also revisit their Gell--Mann and Heisenberg--Weyl scalarization procedures. In the Heisenberg--Weyl setting, the prime-dimensional case already yields the optimal interaction exponent, whereas for composite $K$ the scalar total degree used in their reduction leads to a larger exponent. A~support-sensitive formulation recovers the optimal interaction exponent for every $K$, while the intrinsic argument gives the stronger dependence on the local dimension. As further consequences, we determine the sharp scale, up to factors exponential only in $d$ and $K$, of coefficient $\ell_1$-normalization and unconditionality on the exact-support spaces. This yields dimension-free coefficient sparsification, including sparse generalized-Pauli approximation in Heisenberg--Weyl coordinates, as well as normalization and query bounds for canonical LCU/qubitization constructions.