AI 中文总结
本文证明了局部量子比特系统上无维数Bohnenblust--Hille不等式，采用内蕴方法获得最优指数和常数，并确定了渐近基的精确尺度，推广了相关先前工作。
AI 中文摘要
我们研究了$K$级量子比特系统上局部算子的无维数Bohnenblust--Hille不等式。对于支撑大小至多为$d$的算子空间，在所有张量积正交算子基上一致地，我们证明了具有最优指数$2d/(d+1)$和常数阶为$O(\sqrt K)^d$的Bohnenblust--Hille不等式。我们的证明是内蕴的，基于单点标量化和块横截解耦，不依赖于对循环群的标量化或Remez型论证。我们通过证明渐近指数Bohnenblust--Hille基满足$ cK^{1/4}\le \beta(K)\le C\sqrt K $（其中$c,C>0$为普适常数）来补充上界。在这个意义上，本工作可视为从互补的内蕴视角延续Slote--Volberg--Zhang的工作线。我们还重新审视了他们的Gell--Mann和Heisenberg--Weyl标量化程序。在Heisenberg--Weyl设置中，素数维情形已经产生最优交互指数，而对于复合$K$，他们约化中使用的标量总次数导致更大的指数。一个支撑敏感的表述为每个$K$恢复了最优交互指数，而内蕴论证给出了对局部维数的更强依赖。作为进一步的结果，我们确定了在仅关于$d$和$K$的指数因子意义下的精确尺度，即精确支撑空间上的系数$\ell_1$-归一化和无条件性。这产生了无维数系数稀疏化，包括Heisenberg--Weyl坐标中的稀疏广义Pauli逼近，以及规范LCU/量子化构造的归一化和查询界。
英文摘要
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 $cK^{1/4}\le β(K)\le C\sqrt K$ 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 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.
CommentsMinor revisions; an explicit conjecture concerning the asymptotic order of $β(K)$ was removed and replaced by an open question
