A universal framework for entanglement detection under group symmetry

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 $(\overlineπ_A\otimes π_B)$-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 $\mathcal{L}$ is decomposable if and only if $\mathcal{L}$ is completely positive or completely copositive. We then apply these results to prove that all PPT quantum channels of the form $$Φ(ρ)=a\frac{\text{Tr}(ρ)}{d}\text{Id}_d+ bρ+cρ^T+(1-a-b-c)\text{diag}(ρ)$$ are entanglement-breaking, and that all A-BC PPT $(U\otimes \overline{U}\otimes U)$-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 $(U\otimes U\otimes U)$-invariant tripartite Werner states [EW01] and resolves some open questions raised in [COS18].