Bell-network states are a class of entangled states of the geometry that satisfy an area-law for the entanglement entropy in a limit of large spins and are automorphism-invariant, for arbitrary graphs. We present a comprehensive analysis of the effective geometry of Bell-network states on a dipole graph. Our main goal is to provide a detailed characterization of the quantum geometry of a class of diffeomorphism-invariant, area-law states representing homogeneous and isotropic configurations in loop quantum gravity, which may be explored as boundary states for the dynamics of the theory. We found that the average geometry at each node in the dipole graph does not match that of a flat tetrahedron. Instead, the expected values of the geometric observables satisfy relations that are characteristic of spherical tetrahedra. The mean geometry is accompanied by fluctuations with considerable relative dispersion for the dihedral angle, and perfectly correlated for the two nodes.

中文翻译：

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偶极子图上 Bell 网络状态的有效几何


贝尔网络状态是几何的一类纠缠状态，它们在大自旋的极限内满足纠缠熵的面积定律，并且对于任意图是自同态不变的。我们全面分析了偶极子图上 Bell 网络状态的有效几何结构。我们的主要目标是提供一类微分同构不变、面积定律状态的详细表征，这些态态代表环量子引力中的齐次和各向同性构型，可以作为理论动力学的边界态进行探索。我们发现偶极子图中每个节点的平均几何形状与平面四面体的几何形状不匹配。相反，几何可观察对象的期望值满足球面四面体特征的关系。平均几何形状伴随着波动，二面角具有相当大的相对离散度，并且两个节点完全相关。