Fun with replicas: tripartitions in tensor networks and gravity

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Abstract
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 dAdB. 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 Gn(A : B : C) for integer n ≥ 2 that generalize G = G2. In holography, the computation of Gn(A : B : C) for n > 2 spontaneously breaks part of a ℤn × ℤn replica symmetry. This prevents any naive application of the Lewkowycz-Maldacena trick in a hypothetical analytic continuation to n = 1.

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