Algorithmic procedure for minimal-term convex decomposition of channels into generalized extreme components

Develop an analytical formula or algorithmic procedure to compute a convex decomposition of an arbitrary completely positive trace-preserving map into generalized extreme channels, ideally using at most m terms as posited by Ruskai’s conjecture, and determine conditions ensuring the minimal number of components does not exceed m.

Background

While the existence of convex partitions of channels into smaller components is guaranteed, determining an explicit decomposition with a bounded number of terms (not exceeding m) is nontrivial. The authors rely on numerical optimization to find approximate decompositions, noting that no general analytical or algorithmic procedure is known.

This problem is closely related to the broader Ruskai conjecture but distinctly focuses on constructive algorithms and minimality of the number of terms, rather than mere existence.