Robust and fragile entanglement of three qubits: Relation to permutation symmetry

The robustness of entanglement for the set of eight total spin states of three qubits is characterized and related to their permutation symmetries. These states fall into four sets of two entangled states, differing in their patterns of robustness to entanglement when one of the states is lost. Their entanglement measures are shown to contain certain permutation symmetries of the spin states and their corresponding marginal two-particle states. The eight entangled states are also found to be eigenstates of the Heisenberg Hamiltonian, σ→_A⋅σ→_B+σ→_A⋅σ→_C+σ→_B⋅σ→_C/2. This allows the entanglement to be tuned in a systematic way by adjusting the symmetries of a general three-qubit Hamiltonian. The necessary and sufficient conditions for separability of a general three-particle state in terms of (a) three one-particle states and (b) one one-particle and one two-particle state are also given.