The role of phases in detecting three-qubit entanglement

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 $7$.

25 pages; to appear in Journal of Mathematical Physics

References in corpus (15)

• Entanglement detection
• Quantum Open System Theory: Bipartite Aspects
• Genuinely Multipartite Concurrence of N-qubit X-matrices
• Entanglement criteria and full separability of multi-qubit quantum states
• Optimal Detection of Entanglement in GHZ States
• Qubit-qudit states with positive partial transpose
• Generalized X states of N qubits and their symmetries
• Construction of multi-qubit optimal genuine entanglement witnesses
• Three-qubit entanglement witnesses with the full spanning properties
• Various notions of positivity for bi-linear maps and applications to tri-partite entanglement
• Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose
• Boundary of the set of separable states
• Separability of three qubit Greenberger-Horne-Zeilinger diagonal states
• Product vectors in the ranges of multi-partite states with positive partial transposes and permanents of matrices
• Maximally genuine multipartite entangled mixed X-states of N-qubits

Cited by in corpus (5)

• Alternatives of entanglement depth and metrological entanglement criteria
• On the convex cones arising from classifications of partial entanglement in the three qubit system
• Three qubit separable states of length ten with unique decompositions
• Indecomposable exposed positive bi-linear maps between two by two matrices
• Separability of multi-qubit states in terms of diagonal and anti-diagonal entries