Continuity of multipartite entanglement spreads

Determine whether the definition of an entanglement spread can be suitably refined so that all resulting spreads are continuous, including at states where the defining trace-invariant vanishes, by resolving the possible failures of continuity arising from the geometry of the zero sets of the relevant trace-invariants.

Background

The paper defines an entanglement spread from a pair of colored graphs whose associated trace-invariants satisfy a domination inequality. The spread is expressed as a logarithmic ratio, with value plus infinity when the denominator vanishes. Although this definition guarantees positivity and LO-monotonicity, the authors point out that continuity is not automatic at zeros of the denominator.

The unresolved issue concerns whether all pathological discontinuities can be eliminated by modifying the definition. The problem depends on the geometry and mutual relations of the projective algebraic subvarieties defined by the equations corresponding to the two trace-invariants.

References

Whether or not it is possible to prevent all such failures of continuity by suitably refining Def.~\ref{def:entanglement_spread} is a corny question that hinges on the respective geometries of the projective subvarieties defined by the two polynomial equations $\tr_G = 0$ and $\tr_H = 0$, as well as their relations. We leave this question open for future work.

— Multipartite entanglement spreads  (2610.12271 - Carrozza et al., 8 Oct 2026) in Remark following Proposition 2.2, Section 2.1 (Definition and LO-monotonicity of entanglement spreads)