Complete measured-chi-squared ordering versus signed lifts

Determine whether complete unsmoothed measured-chi-squared ordering between two quantum channels implies the existence of a complete variance-dominating signed lift in general.

Background

The paper establishes a hierarchy in which complete variance-dominating signed lifts imply complete measured-χ2\chi^2 ordering, while the converse is proved only in special settings. Specifically, the converse is settled for unital qubit channel pairs and for source channels with full output span. The qutrit-based examples satisfy both properties, whereas the activating examples fail both, so the general converse remains unresolved.

References

Whether complete unsmoothed measured-$\chi2$ ordering implies a complete signed lift remains open in general.

— Quantum estimation, channel orders, and private capacity  (2610.03706 - Hirche, 2 Oct 2026) in Section 6, Discussion (paragraph beginning “Whether complete unsmoothed measured-$\chi^2$ ordering…”)