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Krenn-Gu Conjecture for Sparse Graphs

Authors L. Sunil Chandran , Rishikesh Gajjala , Abraham M. Illickan

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Author Details

L. Sunil Chandran
  • Indian Institute of Science, Bengaluru, India
Rishikesh Gajjala
  • Indian Institute of Science, Bengaluru, India
Abraham M. Illickan
  • University of California, Irvine, CA, USA

Funding

  • Chandran, L. Sunil: Supported by SERB Core Research Grant CRG/2022/006770: "Bridging Quantum Physics with Theoretical Computer Science and Graph Theory".

Cite As Get BibTex

L. Sunil Chandran, Rishikesh Gajjala, and Abraham M. Illickan. Krenn-Gu Conjecture for Sparse Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.41

Abstract

Greenberger–Horne–Zeilinger (GHZ) states are quantum states involving at least three entangled particles. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. Krenn, Gu and Zeilinger discovered a correspondence between a large class of quantum optical experiments which produce GHZ states and edge-weighted edge-coloured multi-graphs with some special properties called the GHZ graphs. On such GHZ graphs, a graph parameter called dimension can be defined, which is the same as the dimension of the GHZ state produced by the corresponding experiment. Krenn and Gu conjectured that the dimension of any GHZ graph with more than 4 vertices is at most 2. An affirmative resolution of the Krenn-Gu conjecture has implications for quantum resource theory. Moreover, this would save huge computational resources used for finding experiments which lead to higher dimensional GHZ states. On the other hand, the construction of a GHZ graph on a large number of vertices with a high dimension would lead to breakthrough results.
In this paper, we study the existence of GHZ graphs from the perspective of the Krenn-Gu conjecture and show that the conjecture is true for graphs of vertex connectivity at most 2 and for cubic graphs. We also show that the minimal counterexample to the conjecture should be 4-connected. Such information could be of great help in the search for GHZ graphs using existing tools like PyTheus. While the impact of the work is in quantum physics, the techniques in this paper are purely combinatorial, and no background in quantum physics is required to understand them.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
Keywords
  • Graph colourings
  • Perfect matchings
  • Quantum Physics

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