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 $2d/(d+1)$ whose asymptotic exponential base in the interaction order is of order $O(\sqrt{K})$. 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 $c{K}^{1/4}\le \beta (K)\le C\sqrt{K}$ 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 at the level of the asymptotic exponential base. As further consequences, we determine the sharp scale, up to factors exponential in $d$ with base depending only on $K$, 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.

Minor revisions; an explicit conjecture concerning the asymptotic order of $\beta (K)$ was removed and replaced by an open question