Suitable normalization condition for arbitrary multipartite partition rank

Determine a suitable normalization condition for the semidefinite-programming relaxation of arbitrary partition rank for systems with more than three particles.

Background

The paper reformulates approximation by partition-rank states as an optimization over lifted product states and then relaxes the separability constraint to positive partial transpose (PPT). For three-party partition-rank decompositions, an additional trace constraint can be imposed because every such decomposition can be orthogonalized. The authors explain that the analogous constraint is immediate for slice rank, but not for general partition rank when the number of particles exceeds three.

A suitable condition would allow the PPT-based semidefinite program to remain a faithful relaxation without introducing spurious lifted states. The problem is explicitly left unresolved for arbitrary multipartite partition rank.

References

For arbitrary partition rank of $N>3$ particles, it remains open to find a similar suitable condition.

— Quantifying the dimensionality of multiparticle entanglement via partition rank  (2608.23399 - Denker et al., 24 Aug 2026) in Appendix B, subsection “Semidefinite program,” subsection “SDP relaxation”