Entanglement of Four-Qubit Rank-$2$ Mixed States

TL;DR

This paper calculates entanglement measures for specific four-qubit mixed states combining maximally entangled states and W states, providing insights into their quantum entanglement properties.

Contribution

It introduces explicit calculations of entanglement measures for three rank-two mixed states in four-qubit systems, expanding understanding of their entanglement structure.

Findings

Computed ${ m f F}^{(4)}_j$ for the mixed states.

Derived linear monotones ${ m f G}^{(4)}_j$ for these states.

Discussed potential applications of these entanglement measures.

Abstract

It is known that there are three maximally entangled states $\ket{\Phi_1} = (\ket{0000} + \ket{1111}) / \sqrt{2}$, $\ket{\Phi_2} = (\sqrt{2} \ket{1111} + \ket{1000} + \ket{0100} + \ket{0010} + \ket{0001}) / \sqrt{6}$, and $\ket{\Phi_3} = (\ket{1111} + \ket{1100} + \ket{0010} + \ket{0001}) / 2$ in four-qubit system. It is also known that there are three independent measures ${\cal F}^{(4)}_j \hspace{.2cm} (j=1,2,3)$ for true four-way quantum entanglement in the same system. In this paper we compute ${\cal F}^{(4)}_j$ and their corresponding linear monotones ${\cal G}^{(4)}_j$ for three rank-two mixed states $\rho_j = p \ket{\Phi_j}\bra{\Phi_j} + (1 - p) \ket{\mbox{W}_4} \bra{\mbox{W}_4}$, where $\ket{\mbox{W}_4} = (\ket{0111} + \ket{1011} + \ket{1101} + \ket{1110}) / 2$. We discuss the possible applications of our results briefly.