扭曲卷积恒等式与有限量子对数的幽灵 $r$-SIC
The twisted convolution identity and ghost $r$-SICs from finite quantum dilogarithms
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中文总结 AI 辅助
该研究将Radchenko和Wheeler的秩-1扭曲卷积恒等式证明推广至所有秩-r容许元组，并建立模量子对数与Shintani-Faddeev模余循环的字典，从而在特定条件下证明幽灵r-SIC的存在，并在Stark猜想下与厄米r-SIC Galois共轭。
中文摘要 AI 辅助
Radchenko 和 Wheeler (RW) 最近证明了模量子对数的实二次特殊值的有限五边形关系，并利用它建立了当前作者所猜想的秩-$1$ 扭曲卷积恒等式。RW 在主情形中给出了一个明确的论证，并指出他们的证明对所有秩-$1$ 容许元组都成立。我们将他们的证明推广到所有秩-$r$ 容许元组，并在各自的论文中提供了模量子对数与 Shintani-Faddeev 模余循环约定之间的显式字典。因此，我们证明了：若 $d,r$ 是正整数，且满足 $r<\frac{d-1}{2}$ 和 $\frac{d^2-1}{r(d-r)} \in \mathbb{Z}$，则存在幽灵 $r$-SIC，即 $\mathbb{C}^d$ 中 $d^2$ 个秩-$r$ 子空间的配置，满足非厄米等弦条件。在 Stark 猜想下，这些配置与称为 $r$-SIC（或秩-$r$ SIC-POVM）的厄米等弦配置是 Galois 共轭的。
英文摘要
Radchenko and Wheeler (RW) recently proved a finite pentagon relation for real quadratic special values of the modular quantum dilogarithm and used it to establish the rank-$1$ twisted convolution identity conjectured by the current authors. RW gave an explicit argument in the principal case and remarked that their proof holds for all rank-$1$ admissible tuples. We extend their proof to all rank-$r$ admissible tuples and provide an explicit dictionary between the modular quantum dilogarithm and the Shintani-Faddeev modular cocycle conventions in the respective papers. Thus, we establish that, if $d,r$ are positive integers such that $r<\frac{d-1}{2}$ and $\frac{d^2-1}{r(d-r)} \in \mathbb{Z}$, then there exist ghost $r$-SICs: i.e., configurations of $d^2$ rank-$r$ subspaces in $\mathbb{C}^d$ that satisfy a non-Hermitian equichordal condition. Under the Stark conjecture, these configurations are Galois conjugate to Hermitian equichordal configurations called $r$-SICs (or rank-$r$ SIC-POVMs).
发表机构
- University of Sydney（悉尼大学）
- Virginia Tech（弗吉尼亚理工大学）
- Phasecraft
- Louisiana State University（路易斯安那州立大学）
机构由 AI 辅助整理，请以论文原文为准。
