Testing the Monogamy Relations via Rank-2 Mixtures

TL;DR

This paper introduces new entanglement measures based on monogamy relations, computes them for specific four-qubit states and a rank-2 mixture, and discusses their properties and limitations.

Contribution

It proposes novel tangle- and negativity-based entanglement measures derived from monogamy relations and computes them explicitly for certain four-qubit states and mixtures.

Findings

The measure t_1 is trivial for the rank-2 mixture, indicating a potential limitation of the monogamy inequality.

The measures n_1 and n_2 are explicitly computed for the mixture, with optimal decompositions derived.

The measure t_2 could not be computed due to the complexity of residual entanglement calculation.

Abstract

We introduce two tangle-based four-party entanglement measures $t_1$ and $t_2$, and two negativity-based 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 $\rho_4 = p \ket{\mbox{GHZ}_4} \bra{\mbox{GHZ}_4} + (1 - p) \ket{\mbox{W}_4} \bra{\mbox{W}_4}$ by finding the corresponding optimal decompositions. It turns out that $t_1 (\rho_4)$ 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 $t_2 (\rho_4)$ due to the difficulty for the calculation of the residual entanglement. The negativity-based measures $n_1 (\rho_4)$ and $n_2 (\rho_4)$ are explicitly computed and the corresponding optimal decompositions are also derived explicitly.