Algorithmic construction of SSA-compatible extreme rays of the subadditivity cone and the N = 6 solution
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1.
California Institute of Technology
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2.
University of California, Davis
Abstract
We compute the set of all extreme rays of the 6-party subadditivity cone that are compatible with strong subadditivity. In total, we identify 208 new (genuine 6-party) orbits, 52 of which violate at least one known holographic entropy inequality. For the remaining 156 orbits, which do not violate any such inequalities, we construct holographic graph models for 150 of them. For the final 6 orbits, it remains an open question whether they are holographic. Consistent with the strong form of the conjecture in [1], 148 of these graph models are trees. However, 2 of the graphs contain a "bulk cycle", leaving open the question of whether equivalent models with tree topology exist, or if these extreme rays are counterexamples to the conjecture. The paper includes a detailed description of the algorithm used for the computation, which is presented in a general framework and can be applied to any situation involving a polyhedral cone defined by a set of linear inequalities and a partial order among them to find extreme rays corresponding to down-sets in this poset.
Copyright and License
© The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Article funded by SCOAP3.
Acknowledgement
We would like to especially thank David Simmons-Duffin for generously providing us super-computing time in the early iterations of the algorithm. We would also like to thank Sergio Hernández-Cuenca and Sridip Pal for useful discussions.
T.H. is supported by the Heising-Simons Foundation “Observational Signatures of Quantum Gravity” collaboration grant 2021 -2817, the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, and the Walter Burke Institute for Theoretical Physics. V.H. has been supported in part by the U.S. Department of Energy grant DE-SC0009999 and by funds from the University of California. M.R. acknowledges support from U.K. Research and Innovation (UKRI) under the U.K. government’s Horizon Europe guarantee (EP/Y00468X/1), and by funds from the University of California during the early stages of this work.
V.H. acknowledges the hospitality of the Kavli Institute for Theoretical Physics (KITP), the Aspen Center for Physics, and the Tsinghua Southeast Asia Center, where part of this work was done. V.H. and M.R. would like to thank the Yukawa Institute for Theoretical Physics for hospitality during the program “Quantum Information, Quantum Matter and Quantum Gravity” during the early stages of this work.
All data is available within the paper, as an ancillary file in the arXiv, and on GitHub [48]. The code used to generate and analyze this data is a combination of a Wolfram Mathematica implementation of the algorithm outlined in this work, and the freely available software Normaliz [30].
For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.
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Additional details
- Heising-Simons Foundation
- 2021 -2817
- United States Department of Energy
- DE-SC0011632
- California Institute of Technology
- Walter Burke Institute for Theoretical Physics -
- United States Department of Energy
- DE-SC0009999
- University of California System
- UK Research and Innovation
- EP/Y00468X/1
- SCOAP3
- Accepted
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2025-04-16
- Available
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2025-06-05Published
- Caltech groups
- Walter Burke Institute for Theoretical Physics, Division of Physics, Mathematics and Astronomy (PMA)
- Publication Status
- Published