A universal framework for entanglement detection under group symmetry - Archive ouverte HAL
Journal Articles Journal of Physics A: Mathematical and Theoretical Year : 2024

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 (π¯¯¯A⊗πB)-invariant quantum states are separable if and only if all extremal unital positive (πB,πA)-covariant maps are decomposable where πA,πB are unitary representations of a compact group and πA is irreducible. Moreover, an extremal unital positive (πB,πA)-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)dIdd+bρ+cρT+(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].

Dates and versions

hal-04685638 , version 1 (03-09-2024)

Identifiers

Cite

Sang-Jun Park, Yeong-Gwang Jung, Jeongeun Park, Sang-Gyun Youn. A universal framework for entanglement detection under group symmetry. Journal of Physics A: Mathematical and Theoretical, 2024, 57, pp.325304. ⟨10.1088/1751-8121/ad6413⟩. ⟨hal-04685638⟩
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