Abstract
One of the most fundamental questions in quantum information theory is PPT-entanglement of quantum states, which is an NP-hard problem in general. In this paper, however, we prove that all PPT (π_A⊗ π_B)-invariant quantum states are separable if and only if all extremal unital positive (π_B,π_A)-covariant maps are decomposable where π_A,π_B are unitary representations of a compact group and π_A is irreducible. Moreover, an extremal unital positive (π_B,π_A)-covariant map L is decomposable if and only if L is completely positive or completely copositive. We then apply these results to prove that all PPT quantum channels of the form Φ(ρ)=aTr(ρ)/dId_d+ bρ+cρ^T+(1-a-b-c)diag(ρ) are entanglement-breaking, and that all A-BC PPT (U⊗ U⊗ U)-invariant tripartite quantum states are A-BC separable. The former strengthens some conclusions in [VW01,KMS20], and the latter provides a strong contrast to the fact that there exist PPT-entangled (U⊗ U⊗ U)-invariant tripartite Werner states [EW01] and resolves some open questions raised in [COS18].