Characterize whether all-partition Rényi multi-entropies distinguish stabilizer states

Determine whether there exists a pair of quantum states whose Rényi multi-entropies are identical for every multipartition, and, if such states exist, construct a minimal example.

Background

The paper exhibits two graph states that are not locally Clifford-equivalent but have identical Rényi multi-entropy spectra for the finest partition, in which every qubit is treated as a separate party.

It remains unresolved whether refining the comparison to the complete collection of Rényi multi-entropies over all multipartitions eliminates this indistinguishability. The authors explicitly ask whether such a pair exists and request a minimal example.

References

Does there exist a pair of states, for which all multi-Rényi entropies over all partitions are identical? If so, what is such a minimal example? We leave this question to future work.

— Rényi Multi-entropy for Stabilizer States: A Finite Field Gauss Sum Problem  (2610.26814 - Czech et al., 19 Oct 2026) in Section 6, “Discussion and outlook”