Identify other quantum divergences yielding spectral points

Determine whether quantum divergences other than the Petz and sandwiched Rényi divergences, when used to define optimized graph information radii, yield spectral points outside the common Petz–sandwiched family.

Background

The proof of spectrality uses three structural properties: an explicit supported minimization with a tower identity, suitable block behavior for disjoint unions and flagged mixtures, and data processing.

For sandwiched Rényi divergences, the paper shows that the optimized graph radii coincide with the Petz family on the proven parameter range, so they do not produce additional spectral points. The unresolved question is whether other divergences possessing analogous properties can enlarge the family.

References

Do other quantum divergences with these three features yield spectral points outside this common family?

— The Lovász number is not alone: a continuum of Rényi points in the entanglement-assisted spectrum of graphs  (2610.20775 - Mghirbi et al., 14 Oct 2026) in Section 6, paragraph “Other divergences”