Testing the monogamy relations via rank-2 mixtures

Abstract

We introduce two tangle-based four-party entanglement measures t 1 and t 2 , and two negativitybased measures n 1 and n 2 , which are derived from the monogamy relations. These measures are computed for three four-qubit maximally entangled and W states explicitly. We also compute these measures for the rank-2 mixture ρ 4 = p|GHZ 4 GHZ 4 |+(1 − p)|W 4 W 4 | by finding the corresponding optimal decompositions. It turns out that t 1 (ρ 4 ) is trivial and the corresponding optimal decomposition is equal to the spectral decomposition. Probably, this triviality is a sign of the fact that the corresponding monogamy inequality is not sufficiently tight. We fail to compute t 2 (ρ 4 ) due to the difficulty for the calculation of the residual entanglement. The negativity-based measures n 1 (ρ 4 ) and n 2 (ρ 4 ) are explicitly computed and the corresponding optimal decompositions are also derived explicitly. arXiv:1607.00135v2 [quant-ph]