Extremal PNCP maps are not finite-copy implementable

Establish that every extremal positive but not completely positive (PNCP) linear map between finite-dimensional matrix algebras admits no finite-copy implementation; equivalently, prove that for any extremal PNCP map Λ, there exists no finite N for which a completely positive N-copy extension Λ_N satisfies Λ_N(ρ^{⊗ N}) = Λ(ρ) for all density operators ρ.

Background

The paper studies when positive but not completely positive maps can be implemented exactly by consuming multiple copies of an input state, introducing the notion of an N-copy extension Λ_N that must be completely positive and satisfy Λ_N(ρ^{⊗ N}) = Λ(ρ). The authors show a necessary-and-sufficient criterion for implementability via positivity of a symmetrized Choi operator and provide examples.

They analyze boundary and extremal positive maps. In two dimensions, any extremal PNCP map is a transposition followed by an invertible CP map, and since transposition itself is not finite-copy implementable, such extremal maps are not implementable with finitely many copies. For higher dimensions, they exhibit the Choi map as an extremal PNCP map that also fails finite-copy implementability by their necessary condition. Motivated by these examples, they propose a general conjecture about all extremal PNCP maps.