AI 中文总结
本研究证明七体绝对最大纠缠态（AME）存在当且仅当局域维数d≥3，通过构造奇数维循环二次相位态、模4余2维的二元-奇维耦合结构，结合已有结论完成所有d≥3时的存在性分类。
AI 中文摘要
我们证明：七个qudits的绝对最大纠缠态存在当且仅当局域维数满足$d\geq 3$。在本工作之前，据我们所知，仅当$d$是除2之外的素数幂，或$d$可表示为已证明存在AME(7,d)态的维数的乘积时，AME(7,d)态被认为存在。由于已证明不存在AME(7,2)态，因此仍需确定所有$d\geq 3$时的存在性。我们为每个奇局域维数构造循环二次相位态，并为每个模4余2的维数提出二元-奇维耦合构造。结合已知的2的幂次情况及AME态的乘积性质，这些构造覆盖了所有局域维数$d\geq 3$。
英文摘要
We prove that an absolutely maximally entangled state of seven qudits exists if and only if the local dimension satisfies $d\geq 3$. Prior to this work, to the best of our knowledge, $\text{AME}(7,d)$ states were known to exist only when $d$ is a prime power other than $2$, or when $d$ can be expressed as a product of dimensions for which existence was already known. Since it has been proved that no $\text{AME}(7,2)$ state exists, it remains to establish existence for all $d\geq 3$. We construct cyclic quadratic-phase states for every odd local dimension and develop a coupled binary--odd-dimensional construction for every dimension congruent to $2$ modulo $4$. Together with the known power-of-two cases and the product property of AME states, these constructions cover every local dimension $d\geq 3$.
