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(√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 cK1/4≤ β(K)≤ C√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 ℓ1-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.
Related
- Polynomial Bohnenblust--Hille Bounds and Hardy--Littlewood Inequalities for Finite Local Systems
- Polynomial growth of Bohnenblust--Hille constants on the Hamming cube
- Asymptotic Contractivity of Bohnenblust-Hille Constants with Bounded Monomial Support
- Polynomial Bohnenblust--Hille bounds for product of cyclic groups
- Polynomial growth of complex polynomial Bohnenblust--Hille constants
BibTeX
@misc{defant2026intrinsic,
title = {{Intrinsic Bohnenblust--Hille Inequalities for Local Qudit Systems}},
author = {Andreas Defant and Daniel Galicer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.13084},
eprint = {2609.13084},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13084},
}
RIS
TY - UNPB
TI - Intrinsic Bohnenblust--Hille Inequalities for Local Qudit Systems
AU - Andreas Defant
AU - Daniel Galicer
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.13084
UR - https://arxiv.org/abs/2609.13084
AB - 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(√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 cK^1/4≤ β(K)≤ C√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 ℓ_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.
ER -