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The Quantum Entropy Cone of Hypergraphs

by Ning Bao, Newton Cheng, Sergio Hernández-Cuenca, Vincent P. Su

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Submission summary

Authors (as Contributors): Newton Cheng · Sergio Hernandez-Cuenca
Submission information
Arxiv Link: (pdf)
Date accepted: 2020-10-19
Date submitted: 2020-04-17 02:00
Submitted by: Cheng, Newton
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


In this work, we generalize the graph-theoretic techniques used for the holographic entropy cone to study hypergraphs and their analogously-defined entropy cone. This allows us to develop a framework to efficiently compute entropies and prove inequalities satisfied by hypergraphs. In doing so, we discover a class of quantum entropy vectors which reach beyond those of holographic states and obey constraints intimately related to the ones obeyed by stabilizer states and linear ranks. We show that, at least up to 4 parties, the hypergraph cone is identical to the stabilizer entropy cone, thus demonstrating that the hypergraph framework is broadly applicable to the study of entanglement entropy. We conjecture that this equality continues to hold for higher party numbers and report on partial progress on this direction. To physically motivate this conjectured equivalence, we also propose a plausible method inspired by tensor networks to construct a quantum state from a given hypergraph such that their entropy vectors match.

Published as SciPost Phys. 9, 067 (2020)

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2020-10-13 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2002.05317v1, delivered 2020-10-13, doi: 10.21468/SciPost.Report.2077


1. Clear discussion of context and review of previous related literature on the holographic entropy cone etc; section 2 sets up the required definitions related to graphs and so on.
2. Main results are laid out as theorems and proofs, making manifest any underlying assumptions.
3. Concluding discussion section explains additional conjectures postulated by the authors and also relates the work to higher derivative gravity and non-holographic bits.


1. While the authors link their results with previous (technical) literature, they could make a stronger case for the applications and importance of hypergraphs and their entropy cone. The results are quite technical - very accessible to those who are following the quantum information literature closely, but the overall importance of these results may be underemphasised.


This is a strong paper within a field of considerable current interest to both the quantum information community and the holography (spacetime reconstruction) community. The authors present their technical results quite well - good introductory and review material, clear statements of theorems and proofs - but they could have given more discussion about the potential wider implications of these results.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2020-7-28 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2002.05317v1, delivered 2020-07-27, doi: 10.21468/SciPost.Report.1864


1- Original idea, generalizing successful techniques in a way that could prove useful to two fields (Quantum information and Quantum gravity).

2- Clear statement of past work and new results


1- Leaves unproven some big, seemingly provable conjectures (such as containment of hypergraph cone in stabilizer cone for all number of parties)


This paper presents a novel way of thinking about GHZ entanglement, using graph theoretic techniques. This opens new possibilities, potentially allowing the well-developed field of graph theory to help fully classify entangled states. I recommend this paper be published as is, no big revisions necessary.

It would be exciting if there were a graph theoretic way to understand all forms of entangled states. It would also be exciting if this were provably not possible.

It is my understanding that a conjecture from this paper was later proven by That is satisfying, because as explained in this paper under review, it seemed like this conjecture should be true!

Requested changes

Is good as is.

  • validity: high
  • significance: good
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: perfect

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