Simpler criteria for finite-copy implementability

Develop simpler necessary or sufficient conditions to decide whether a given positive map Λ admits a completely positive N-copy extension Λ_N for some finite N, beyond the eigenvalue-based characterization via the symmetrized Choi operator and the specific sufficient and necessary conditions provided in the paper.

Background

The authors provide a full characterization: Λ is N-copy implementable if and only if the symmetrized N-copy extension Λ_N^sym is completely positive, which reduces to checking the minimum eigenvalue of a specific operator. They also derive two sufficient and one necessary condition to facilitate verification.

Despite these results, they explicitly state that finding simpler criteria than those currently available remains an open direction, indicating a desire for more tractable tests or characterizations for practical use.