Contextuality threshold and measurement contextuality of the 24-cell subtheory

Determine whether incorporating the 24-cell subtheory’s further operational equivalences lowers the preparation-contextuality threshold below depolarising strength 2/3, and determine whether the 24-cell subtheory is measurement contextual.

Background

The paper studies the 24-cell subtheory of the four-dimensional hyperbit, consisting of 24 pure states from three mutually unbiased frames, the associated 12 axis questions, and their mixtures. A regular-hexagon configuration inside the 24-cell establishes preparation contextuality, while a depolarising map shows contextuality for depolarising strength greater than 2/3. The value 2/3 is shown to be tight for the hexagon considered in isolation.

The unresolved issue is whether the complete set of operational equivalences of the full 24-cell subtheory imposes stronger constraints than those used for the hexagon, thereby lowering the contextuality threshold. The paper also leaves unresolved whether the same subtheory exhibits measurement contextuality. Both questions are identified as linear-programming problems.

References

The 24-cell subtheory is therefore preparation contextual at least for $\lambda>2/3$; whether its further equivalences lower the threshold, and whether it is measurement contextual, we leave open, and both are linear programs.

— Why three? A two-level system with four mutually unbiased questions  (2609.10078 - Hance, 9 Sep 2026) in Section “Polytopes, quasiprobabilities and contextuality,” immediately following Theorem 24contextual

Whether the 24-cell's contextuality is a resource for the two-qubit states and measurements that carry it we leave open.

— Why three? A two-level system with four mutually unbiased questions  (2609.10078 - Hance, 9 Sep 2026) in Discussion, final paragraph before “Acknowledgments”