Fun with replicas: tripartitions in tensor networks and gravity

A bstract We analyse a simple correlation measure for tripartite pure states that we call G ( A : B : C ). The quantity is symmetric with respect to the subsystems A , B , C , invariant under local unitaries, and is bounded from above by log d A d B . For random tensor network states, we prove that G ( A : B : C ) is equal to the size of the minimal tripartition of the tensor network, i.e., the logarithmic bond dimension of the smallest cut that partitions the network into three components with A , B , and C . We argue that for holographic states with a fixed spatial geometry, G ( A : B : C ) is similarly computed by the minimal area tripartition. For general holographic states, G ( A : B : C ) is determined by the minimal area tripartition in a backreacted geometry, but a smoothed version is equal to the minimal tripartition in an unbackreacted geometry at leading order. We briefly discuss a natural family of quantities G n ( A : B : C ) for integer n ≥ 2 that generalize G = G 2 . In holography, the computation of G n ( A : B : C ) for n > 2 spontaneously breaks part of a ℤ n × ℤ n replica symmetry. This prevents any naive application of the Lewkowycz-Maldacena trick in a hypothetical analytic continuation to n = 1.