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PreprintCC BY 4.0arXiv

Relative entropy of entanglement and tripartite minimal surface

  • Takato Mori
  • Beni Yoshida
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Abstract

We propose a holographic dual of the relative entropy of entanglement for a boundary tripartition A:B:C. Our proposal is that, at the leading order, ER(ρAB) = 1/4GN[Area(Γmin) − Area(γAB)] notag where Γmin is the minimal bulk tripartition surface and γAB is the minimal surface homologous to AB. For a general random tensor network, we prove the corresponding upper bound by constructing a separable state associated with Γmin and evaluating its relative entropy. We also establish the matching lower bound rigorously for networks of one, two, and three Haar random tensors by bounding the maximal tripartite product-state overlap. In appropriate geometries, the minimal tripartition surface can form a nontrivial tri-junction resembling the Mercedes logo. The proof proceeds by sequential optimization over candidate product states, which has a natural geometric interpretation as a local search for the minimal tripartition surface.

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Cite

BibTeXRIS
BibTeX
@misc{mori2026relative,
  title = {{Relative entropy of entanglement and tripartite minimal surface}},
  author = {Takato Mori and Beni Yoshida},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38815},
  eprint = {2609.38815},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38815},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Relative entropy of entanglement and tripartite minimal surface
AU  - Takato Mori
AU  - Beni Yoshida
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.38815
UR  - https://arxiv.org/abs/2609.38815
AB  - We propose a holographic dual of the relative entropy of entanglement for a boundary tripartition A:B:C. Our proposal is that, at the leading order, E_R(ρ_AB) = 1/4G_N[Area(Γ_min) − Area(γ_AB)] notag where Γ_min is the minimal bulk tripartition surface and γ_AB is the minimal surface homologous to AB. For a general random tensor network, we prove the corresponding upper bound by constructing a separable state associated with Γ_min and evaluating its relative entropy. We also establish the matching lower bound rigorously for networks of one, two, and three Haar random tensors by bounding the maximal tripartite product-state overlap. In appropriate geometries, the minimal tripartition surface can form a nontrivial tri-junction resembling the Mercedes logo. The proof proceeds by sequential optimization over candidate product states, which has a natural geometric interpretation as a local search for the minimal tripartition surface.
ER  -