Analytically computable tangle for three-qubit mixed states

We present a new tripartite entanglement measure for three-qubit mixed states. The new measure t_r(ρ), which we refer to as the r-tangle, is given as a kind of the tangle, but has a feature which the tangle does not have; if we can derive an analytical form of t_r(ρ) for a three-qubit mixed state ρ, we can also derive t_r(ρ') analytically for any states ρ' which are SLOCC-equivalent to the state ρ. The concurrence of two-qubit states also satisfies the feature, but the tangle does not. These facts imply that the r-tangle t_r is the appropriate three-partite counterpart of the concurrence. We also derive an analytical form of the r-tangle t_r for mixtures of a generalized GHZ state and a generalized W state, and hence for all states which are SLOCC-equivalent to them.