## 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 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.

Original language | English |
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Article number | 8 |

Journal | Journal of High Energy Physics |

Volume | 2023 |

Issue number | 5 |

Number of pages | 30 |

ISSN | 1126-6708 |

DOIs | |

Publication status | Published - 2023 |

### Bibliographical note

Publisher Copyright:© 2023, The Author(s).

## Keywords

- AdS-CFT Correspondence
- Black Holes in String Theory