Complete inequality system for the qubit-qutrit qutrit marginal polytope

Determine a concise and complete inequality system describing the spectral marginal polytope of the qutrit marginal in the qubit-qutrit system, beyond the necessary Ky Fan bounds derived in the paper.

Background

The paper identifies the support of the qutrit marginal density with the corresponding one-marginal spectral polytope and derives elementary Ky Fan bounds, including bounds on the largest and smallest qutrit eigenvalues. However, it explicitly notes that these bounds may not be sufficient to characterize the polytope.

A complete description would connect the walls arising from subset sums in the spline formulas with a minimal generating set of Klyachko-type inequalities. The problem is therefore to replace the partial necessary conditions with a concise, irredundant, and sufficient inequality description for the qutrit marginal spectrum in the qubit-qutrit setting.

References

Thus, a valuable future direction would be to systematically map the walls appearing in the spline formulas (which originate from subset sums of the global eigenvalues) to a minimal generating set of Klyachko-type inequalities. In particular, for the qutrit marginal in the qubit-qutrit system, the elementary Ky Fan bounds derived here provide only necessary conditions and may not be sufficient; finding a concise and complete inequality system for this marginal polytope remains an important open problem.

— Marginal spectral distributions on regular bipartite unitary orbits  (2608.30370 - Zhang, 31 Aug 2026) in Section 6, Concluding remarks, subsection “Explicit descriptions of marginal polytopes”