The role of phases in detecting three-qubit entanglement

TL;DR

This paper introduces new criteria based on phases of anti-diagonal entries to detect entanglement in three-qubit states, including PPT entanglement, improving understanding of separability conditions.

Contribution

It reveals the crucial role of phases in anti-diagonal entries for separability and provides criteria that detect PPT entanglement with nonzero volume.

Findings

Phases determine separability in specific three-qubit states.

Criteria detect PPT entanglement with nonzero volume.

Characterization of separability in low-rank states using phases.

Abstract

We propose separability criteria for three-qubit states in terms of diagonal and anti-diagonal entries to detect entanglement with positive partial transposes. We report that the phases, that is, the angular parts of anti-diagonal entries, play a crucial role in determining whether a given three-qubit state is separable or entangled, and they must obey even an identity for separability in some cases. These criteria are strong enough to detect PPT (positive partial transpose) entanglement with nonzero volume. In several cases when all the entries are zero except for diagonal and anti-diagonal entries, we characterize separability using phases. These include the cases when anti-diagonal entries of such states share a common magnitude, and when ranks are less than or equal to six. We also compute the lengths of rank six cases, and find three-qubit separable states with lengths $8$ whose maximum ranks of partial transposes are $7$.