Fun with replicas: tripartitions in tensor networks and gravity

Geoff Penington; Michael Walter; Freek Witteveen

doi:10.1007/JHEP05(2023)008

paper

Fun with replicas: tripartitions in tensor networks and gravity

arXiv:2211.16045 · doi:10.1007/JHEP05(2023)008

Abstract

We introduce a new 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 $lo g d_{A} d_{B}$ . 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 $n \geq 2$ that generalize $G = G_{2}$ . In holography, the computation of $G_{n} (A : B : C)$ for $n > 2$ spontaneously breaks part of a $Z_{n} \times Z_{n}$ replica symmetry. This prevents any naive application of the Lewkowycz-Maldacena trick in a hypothetical analytic continuation to $n = 1$ .

arXiv comment28 pages, 10 figures

Fun with replicas: tripartitions in tensor networks and gravity

Abstract

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