Characterization of regions where ordinary Bargmann invariants are complete

Characterize the subsets of bipartite-state space on which ordinary marginal-word Bargmann invariants completely separate local-unitary orbits, including subsets defined by nondegenerate marginal spectra.

Background

The paper demonstrates that ordinary marginal-word invariants fail to separate local-unitary orbits on the locally maximally mixed sector and also provides a counterexample with nondegenerate, non-maximally mixed marginals. Thus, nondegeneracy alone does not settle the issue. The authors leave unresolved a precise characterization of state-space regions where the ordinary invariant family nevertheless becomes complete.

References

Several problems remain open. A finite separating family for generic qubit--qutrit mixed states is not identified here; the generic orbit space has real dimension $24$, but a complete separating set must also resolve nongeneric strata. It would be useful to characterize the subsets of state space on which ordinary marginal-word invariants remain complete, for instance under nondegeneracy assumptions on the marginal spectra.

— Bargmann Invariants Are Insufficient for Complete Local-Unitary Orbit Discrimination  (2609.11090 - Zhang et al., 10 Sep 2026) in Section 7, Conclusions