We construct a large family of Planar Maximally Entangled (PME) states which are a wider class of multi-partite entangled states than Absolutely Maximally Entangled (AME) states. These are states in which any half of the qudits are in a maximally mixed state, provided that they form a connected subset. We show that in contrast to AMEs, PMEs are easier to find and there are various PMEs for any even number of qudits. In particular, while it is known that no AME state of four qubits exists, we show that there are two distinct multi-parameter classes of four qubit PMEs. We also give explicit families of PMEs for any even number of particles and for any dimension. We also briefly mention the relevance of connectivity and underlying geometry to definition of a PME state. Here we have considered the simplest case of a one dimensional graph.

中文翻译：

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平面最大纠缠态

我们构造了一个大系列的平面最大纠缠（PME）状态，它比绝对最大纠缠（AME）状态具有更广泛的多部分纠缠状态。在这些状态下，只要它们形成一个连接的子集，任何一半的qud都处于最大混合状态。我们证明，与AMEs相比，PME更容易找到，并且对于任何偶数个Qudits都有各种PME。特别地，虽然已知不存在四个量子位的AME状态，但我们显示出四个量子位PME有两个不同的多参数类。我们还为任何偶数个粒子和任何尺寸给出了显式的PME系列。我们还简要提到了连通性和底层几何结构与PME状态定义的相关性。在这里，我们考虑了一维图的最简单情况。