Stability relative to the actual optimal-marginal set

Establish stability bounds that measure the distance of a near-optimal system marginal to the set of system marginals realized by actual optimal protocols, rather than merely to the convex locked marginal set.

Background

The paper proves that a near-optimal zero-type-I-error marginal is close in trace distance to the convex set defined by support and Schur-profile locking. It explicitly notes that this locked set is not known to equal the set of marginals realized by optimal protocols.

Consequently, the existing stability estimate does not establish proximity to an actual optimal marginal. A stronger stability theory relative to the realizable optimal-marginal set remains unresolved.

References

Other open problems include uniform qutrit phase-space separation beyond nine queries, sufficient conditions for a locked marginal to be realizable by an optimal protocol, constraints on optimal measurement effects, and stability relative to the actual optimal-marginal~set.

— Resource Compatibility in Optimal Quantum Symmetry Testing  (2609.38134 - Cao et al., 29 Sep 2026) in Discussion; Supplemental Section I, “Quantitative support and weight locking”