Eliminating the optimization over input measurement assemblages

Determine how to bypass the optimization over input measurement assemblages in the robustness measure \(\mathcal{R}^{\mathrm{JM}}_{\mathrm{IP}}(\mathcal{N})\) for measurement-incompatibility preservability.

Background

The paper defines RIPJM(N)\mathcal{R}^{\mathrm{JM}}_{\mathrm{IP}}(\mathcal{N}) by maximizing a semidefinite-program value over all input measurement assemblages. Because this optimization is bilinear, the authors use a see-saw algorithm that alternates between optimizing the measurement assemblage and the dual variables; this procedure may find, but does not necessarily certify, a globally optimal assemblage.

The open problem is to identify a method that removes or bypasses this measurement-assemblage optimization, potentially by proving a structural reduction analogous to the fact that a maximally entangled input state is optimal when quantifying non-entanglement-breaking channels.

References

In addition, we apply a see-saw algorithm to potentially find out the optimal measurement assemblage in Eq.~eq:robustness JM. The same approach is also applied to finding the most incompatible measurements for achieving the optimal steerability in the level of quantum state. A similar problem, but in a different scheme, has been solved. In quantifying a non-entanglement-breaking channel, the optimal input state is always the maximally entangled state. How can we bypass the optimization of measurement assemblage in Eq.~eq:robustness JM?

— Preservability of Measurement Incompatibility: Purification, Activation, and a No-Go Theorem  (2610.01327 - Wu et al., 1 Oct 2026) in Section Conclusions