AI 中文总结
该研究构造了满足帕塞瓦尔极限的稠密哈密顿量，证明非交换BH常数呈指数增长，还反驳了Bu等人提出的量子FEI猜想，明确了非交换与经典超立方体BH常数的渐近差异。
AI 中文摘要
对于每个$d\geq1$，我们构造了一个范数为1的厄米算子，其泡利展开包含$N(d)=\exp(\Omega(d^2))$项，每项的次数为$d$且幅度为$1/\sqrt{N(d)}$，这是帕塞瓦尔恒等式允许的最大幅度。作为对比，若有界对角算子（等价于布尔立方体上的有界$d$次函数）具有$N(d)$个泡利系数且幅度均为$\Omega(1/\sqrt{N(d)})$，则$N(d)\leq\exp(\widetilde{O}(d^{1.5}))$。该构造意味着非交换Bohnenblust–Hille（BH）常数满足$\mathrm{BH}_{M_2}(d)\geq\exp(\Omega(d))$，结合前人工作证明的上界，确定了$\mathrm{BH}_{M_2}(d)$的渐近增长为指数级。我们的下界还在渐近意义上将$\mathrm{BH}_{M_2}(d)$与（经典）超立方体BH常数$\mathrm{BH}_{\{\pm1\}}(d)$区分开来，已知后者是次指数级的：$\mathrm{BH}_{\{\pm1\}}(d)\leq C^{\sqrt{d\log d}}$。我们的哈密顿量也是幺正的，因此属于Montanaro和Osborne（2010）定义的量子布尔函数，据此反驳了Bu等人（2024）提出的量子傅里叶熵-影响（FEI）猜想，该猜想是Friedgut和Kalai（1996）经典傅里叶熵-影响猜想的自然推广。
英文摘要
For each $d\geq 1$ we construct a norm-1 Hermitian operator whose Pauli expansion contains $N(d)=\exp(Ω(d^2))$ terms, each of degree $d$ and magnitude $1/\sqrt{N(d)}$ - the largest magnitude permitted by Parseval's identity. For comparison, if a bounded diagonal operator (or equivalently, a bounded degree-$d$ function on the Boolean cube) has $N(d)$ Pauli coefficients, all of magnitude $Ω(1/\sqrt{N(d)})$, then $N(d)\leq \exp(\widetilde{O}(d^{1.5}))$. This construction implies the noncommutative Bohnenblust--Hille (BH) constant satisfies $\mathrm{BH}_{M_2}(d)\geq\exp(Ω(d))$. Together with the upper bounds proved in prior work, this settles the asymptotic growth of $\mathrm{BH}_{M_2}(d)$ as exponential. Our lower bound also asymptotically separates $\mathrm{BH}_{M_2}(d)$ from the (classical) hypercube BH constant $\mathrm{BH}_{\{\pm 1\}}(d)$, which in turn is known to be subexponential: $\mathrm{BH}_{\{\pm 1\}}(d)\leq C^{\sqrt{d \log d}}$. Our Hamiltonians are also unitary and thus quantum Boolean functions in the sense of Montanaro and Osborne (2010). As such they refute the quantum Fourier Entropy-Influence conjecture of Bu et al. (2024), a natural generalization of the classical Fourier Entropy-Influence conjecture due to Friedgut and Kalai (1996).
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