Absolutely maximally entangled (AME) states of $k$ qudits (also known as perfect tensors) are quantum states that have maximal entanglement for all possible bipartitions of the sites/parties. We consider the problem of whether such states can be decomposed into a tensor network with a small number of tensors, such that all physical and all auxiliary spaces have the same dimension $D$. We find that certain AME states with $k=6$ can be decomposed into a network with only three 4-leg tensors; we provide concrete solutions for local dimension $D=5$ and higher. Our result implies that certain AME states with six parties can be created with only three two-site unitaries from a product state of three Bell pairs, or equivalently, with six two-site unitaries acting on a product state on six qudits. We also consider the problem for $k=8$, where we find similar tensor network decompositions with six 4-leg tensors.

中文翻译：

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绝对最大纠缠态的张量网络分解

$k$ 量子的绝对最大纠缠 (AME) 状态（也称为完美张量）是对位点/方的所有可能二分都具有最大纠缠的量子态。我们考虑这样的问题：是否可以将这样的状态分解为具有少量张量的张量网络，使得所有物理空间和所有辅助空间具有相同的维度$D$。我们发现某些 $k=6$ 的 AME 状态可以分解为只有三个 4 腿张量的网络；我们为局部尺寸 $D=5$ 及更高提供具体的解决方案。我们的结果表明，具有六个参与方的某些 AME 状态可以仅使用来自三个贝尔对的产品状态的三个两点酉来创建，或者等效地，用六个两点酉作用于六个量子的产品状态。我们还考虑 $k=8$ 的问题，其中我们发现具有六个 4 腿张量的类似张量网络分解。