Holographic Entropy Cone (HEC)

The HEC is a family of polyhedral cones $H_n$ labelled by the number of parties $n \in \mathbb{N}$.
Each $H_n$ can be dually described by either its facet inequalities or its extreme rays.
A complete characterization of $H_n$ requires knowledge of both sets of extremal elements.
This has only been accomplished for $n \leq 5$, and results for $n \geq 6$ are only partial.

Description

This repository collects all known extremal elements of $H_n$ for $n \leq 6$.
Each $H_n$ is invariant under the symmetric group of degree $n+1$.
Files contain one representative element per symmetry orbit.
Elements of $H_n$ obtained as lifts from $H_k$ for $k < n$ are listed first.

Summary

Orbit counts:

n	facets (lifts)	rays (lifts)	status
1	1 (0)	1 (0)	complete
2	1 (0)	1 (1)	complete
3	2 (1)	2 (1)	complete
4	2 (2)	3 (2)	complete
5	8 (3)	19 (3)	complete
6	1877 (11)	4145 (19)	incomplete

Attribution

If you find this data useful for your research, please consider citing the following papers.

• Complete description of $H_n$ for $n \leq 5$ from arXiv:1903.09148:

  @article{n5hec,
      author         = "Hern\'andez Cuenca, Sergio",
      title          = "{The Holographic Entropy Cone for Five Regions}",
      eprint         = "1903.09148",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
      doi            = "10.1103/PhysRevD.100.026004",
      journal        = "Phys. Rev. D",
      volume         = "100",
      number         = "2",
      pages          = "026004",
      year           = "2019",
      note           = "Data available at \url{https://github.com/SergioHC95/Holographic-Entropy-Cone}"
  }

• Partial description of $H_n$ for $n=6$ from arXiv:2309.06296:

  @article{n6hec,
      author         = "Hern\'andez-Cuenca, Sergio and Hubeny, Veronika E. and Jia, Frederic",
      title          = "{Holographic Entropy Inequalities and Multipartite Entanglement}",
      eprint         = "2309.06296",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
      reportNumber   = "MIT-CTP/5610",
      month          = "9",
      year           = "2023",
      note           = "Data available at \url{https://github.com/SergioHC95/Holographic-Entropy-Cone}"
  }