A universal framework for entanglement detection under group symmetry

Abstract

One of the fundamental questions in quantum information theory is determining entanglement of quantum states, which is generally an NP-hard problem. In this paper, 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 ) = a Tr ( ρ ) d Id d + b ρ + c ρ T + ( 1 − a − b − c ) diag ( ρ ) are entanglement-breaking, and that there is no A-BC PPT-entangled ( U ⊗ U ― ⊗ U ) -invariant tripartite quantum state. The former strengthens some conclusions in (Vollbrecht and Werner 2001 Phys. Rev. A 64 062307; Kopszak et al 2020 J. Phys. A: Math. Theor. 53 395306), and the latter resolves some open questions raised in (Collins et al 2018 Linear Algebra Appl. 555 398–411).