Krenn–Gu conjecture on the dimension of GHZ graphs

Prove that every loopless multigraph with more than four vertices has matching index, and hence GHZ-state dimension, at most 2.

Background

The paper studies edge-coloured, edge-weighted graphs whose feasible monochromatic inherited vertex colourings represent the successful outcomes of photonic experiments and whose matching index measures the maximum GHZ-state dimension realizable by a fixed graph. The Krenn–Gu conjecture asserts that no loopless multigraph with more than four vertices can realize dimension greater than 2.

The conjecture remains unresolved in the paper. A proof would establish a fundamental limitation on the considered photonic model, whereas a counterexample would provide a construction of a higher-dimensional multipartite GHZ state. The paper proves several bounds supporting the conjecture, including bounds based on connectivity and degree, but does not settle the general case.

References

This led Krenn and Gu to conjecture that every GHZ graph with more than four vertices has dimension at most $2$.

— Connectivity Bounds for GHZ Graphs  (2610.22190 - Chandran et al., 15 Oct 2026) in Conjecture 1 (Krenn–Gu Conjecture), Section 1, subsection “Problem definition”

They further conjectured that the maximum dimension achievable for an $n>4$ vertex graph is less than $\dfrac{n}{2}$ when bichromatic edges are absent.

— Connectivity Bounds for GHZ Graphs  (2610.22190 - Chandran et al., 15 Oct 2026) in Section 1, subsection “Related work on the conjecture,” paragraph “Absence of bichromatic edges or multi-edges”; summarized in Table 1

For $\kappa(G)=2$, stronger results are already known; for higher connectivity, the optimal dependence on $\kappa(G)$ remains open, and in particular it is not known whether a polynomial bound in $\kappa(G)$ should hold.

— Connectivity Bounds for GHZ Graphs  (2610.22190 - Chandran et al., 15 Oct 2026) in Section 1, subsection “Our Results,” paragraph following Theorem 5 (vertex-connectivity bound)