Three-Tangle of a General Three-Qubit State in the Representation of Majorana Stars

Abstract

Majorana stars, the $2j$ spin coherent states that are orthogonal to a spin-$j$ state, offer a visualization of general quantum states and may disclose deep structures in quantum states and their evolutions. In particular, the genuine tripartite entanglement - the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2 state, is measured by the normalized product of the distance between the Majorana stars. However, the Majorana representation cannot applied to general non-symmetric $n$-qubit states. We show that after a series of SL$(2,\mathbb{C})$ transformations, non-symmetric three-qubit states can be transformed to symmetric three-qubit states, while at the same time the three-tangle is unchanged. Thus the genuine tripartite entanglement of general three-qubit states has the geometric representation of the associated Majorana stars. The symmetrization and hence the Majorana star representation of certain genuine high-order entanglement for more qubits are possible for some special states. In general cases, however, the constraints on the symmetrization may prevent the Majorana star representation of the genuine entanglement.

References in corpus (33)

• Quantum entanglement
• Entanglement of Formation of an Arbitrary State of Two Qubits
• Three qubits can be entangled in two inequivalent ways
• Distributed Entanglement
• Persistent entanglement in arrays of interacting particles
• Four qubits can be entangled in nine different ways
• Generalized Schmidt decomposition and classification of three-quantum-bit states
• Classification of multipartite entangled states by multidimensional determinants
• Multi-particle entanglement and its applications to cryptography
• The polynomial invariants of four qubits
• Operational Families of Entanglement Classes for Symmetric $N$-Qubit States
• Classifying Novel Phases of Spinor Atoms
• The maximally entangled symmetric state in terms of the geometric measure
• All Maximally Entangled Four Qubits States
• Entanglement and symmetry in permutation symmetric states
• On polynomial invariants of several qubits
• Entanglement in the symmetric sector of $n$ qubits
• Entanglement Equivalence of $N$-qubit Symmetric States
• Optimal state for keeping reference frames aligned and the Platonic solids
• Quantum geometric phase in Majorana's stellar representation: Mapping onto a many-body Aharonov-Bohm phase
• Tensor Representation of Spin States
• Quantum metrology at the limit with extremal Majorana constellations
• Classifying Vortices in S=3 Bose-Einstein Condensates
• Quantum-limited Euler angle measurements using anticoherent states
• Representation of Berry phase by the trajectories of Majorana stars
• Inert states of spin-S systems
• Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere
• Multipartite-entanglement monotones and polynomial invariants
• Anticoherence of spin states with point group symmetries
• Majorana representation for mixed states
• An alternative representation for pure symmetric states of qubits and its applications to entanglement classification
• Permutation-invariant monotones for multipartite entanglement characterization
• Geometric Multiaxial Representation of N-qubit Mixed Symmetric Separable States

Cited by in corpus (4)

• Majorana representation of adiabatic and superadiabatic processes in three-level systems
• Genuine tripartite entanglement as a probe of quantum phase transitions in a spin-1 Heisenberg chain with single-ion anisotropy
• Berry phases of higher spins due to internal geometry of Majorana constellation and relation to quantum entanglement
• Entanglement Classification via Single Entanglement Measure