A universal framework for entanglement detection under group symmetry

TL;DR

This paper establishes a universal framework for detecting entanglement in quantum states with group symmetry, linking PPT invariance to separability and decomposability of certain maps, with applications to quantum channels and tripartite states.

Contribution

It provides a new criterion connecting PPT invariance and map decomposability, and applies this to characterize entanglement-breaking channels and symmetric tripartite states.

Findings

All PPT invariant states are separable if extremal covariant maps are decomposable.

Certain PPT quantum channels are entanglement-breaking.

All PPT invariant tripartite states under specific symmetry are separable.

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 $(\overline{\pi}_A\otimes \pi_B)$-invariant quantum states are separable if and only if all extremal unital positive $(\pi_B,\pi_A)$-covariant maps are decomposable where $\pi_A,\pi_B$ are unitary representations of a compact group and $\pi_A$ is irreducible. Moreover, an extremal unital positive $(\pi_B,\pi_A)$-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 $$\Phi(\rho)=a\frac{\text{Tr}(\rho)}{d}\text{Id}_d+ b\rho+c\rho^T+(1-a-b-c)\text{diag}(\rho)$$ 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].