Holographic Entropy Cone

The Holographic Entropy Cone (HEC) is a family of polyhedral cones (H_n), labelled by the number of parties (n). Each cone has a dual description in terms of facet inequalities and extreme rays. The cones are known completely for (n \leq 5); the (n=6) data in this repository is partial.

Data

The repository contains one representative from each known full-symmetry orbit under (S_{n+1}). Data files live under data/n=*/:

• facets.json contains facet-inequality representatives.
• rays.json contains extreme-ray representatives.
• graphs.json contains exact graph realizations for the listed rays.
• contractions.json contains contraction-map certificates for the listed facets.

Rows use primitive integer coefficients in cardinality-then-lexicographic subset order. Facet and ray files list lifts from lower-party cones first. The remaining facet rows—historical and newly added—are mixed and sorted row-lexicographically. The contraction record at each position proves the facet at that position.

Graphs contain only edges and weights. Contractions contain only lhs, rhs, and images.

For n=6, contractions.json is a manifest for the aligned contractions.packbits.zst stream. The stream stores one binary contraction table per facet, in facet-file order, using little-endian row-major packbits. The manifest records dimensions, hashes, sizes, and ordering information so the complete stream can be checked without a solver.

Python package

The hec package under src/hec/ provides:

• contraction search and certificate checking;
• facet and ray rank checks;
• exact graph realization and verification;
• loading and validating the repository data;
• symmetry actions, canonicalization, and shared serialization utilities.

Install the locked dependencies in an environment on the operating system where the code will run:

uv sync --locked --dev
uv run python -m hec.checks database
uv run ruff check .

The contraction solver requires a PySAT build exposing Kissat and its raw C symbols. The tracked Cython sources are built automatically on first use when the local extensions are absent. On Windows, run the installation and checks from an x64 Visual C++ environment such as the “x64 Native Tools Command Prompt for VS”.

Set worker counts explicitly when desired:

HEC_WORKERS=8 uv run python examples/find_ineq_contractions.py
HEC_GRAPH_WORKERS=4 uv run python examples/find_ray_graphs.py
HEC_CHECK_WORKERS=16 uv run python -m hec.checks facets

The corresponding PowerShell form is:

$env:HEC_WORKERS = "8"
uv run python examples\find_ineq_contractions.py
$env:HEC_GRAPH_WORKERS = "4"
uv run python examples\find_ray_graphs.py
$env:HEC_CHECK_WORKERS = "16"
uv run python -m hec.checks facets

Summary

The table gives orbit-representative counts and the number of distinct images of those representatives. Stabilizer-related repeats within an orbit are counted once.

n	facet reps (lifts)	facet images	ray reps (lifts)	ray images	status
1	1 (0)	1	1 (0)	1	complete
2	1 (0)	3	1 (1)	3	complete
3	2 (1)	7	2 (1)	7	complete
4	2 (2)	20	3 (2)	20	complete
5	8 (3)	372	19 (3)	2,267	complete
6	59,191 (11)	293,632,283	4,161 (19)	15,455,986	incomplete

Data updates:

• 2026-07-30: computed and verified the complete n=6 facet-image total, 287,558,243, for the 57,940 stored representatives.
• 2026-07-28: added 52,211 retained new facet representatives and their proved contraction certificates. Historical representative values were preserved, and all n=1 through n=6 facet files were sorted with lifts first and the remaining old/new rows mixed in row-lexicographic order. The 57,940 aligned n=6 contraction tables are stored as a 251.7 MiB zstd packbits stream (1.83 GiB raw), with exact dimensions and SHA-256 metadata.

Attribution

If you use this data, 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 https://github.com/SergioHC95/Holographic-Entropy-Cone"
  }

• Partial description of (H_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 https://github.com/SergioHC95/Holographic-Entropy-Cone"
  }