Relative entropy of entanglement for multipartite mixed states: Permutation-invariant states and Dür states

We prove conjectures on the relative entropy of entanglement (REE) for two families of multipartite qubit states. Thus, analytical expressions of the REE for these families of states can be given. The first family of states is composed of mixtures of some permutation-invariant multiqubit states. The results generalized to multiqudit states are also shown to hold. The second family of states contains Dür’s bound entangled states. Along the way, we discuss the relation of the REE to two other measures: robustness of entanglement and the geometric measure of entanglement, slightly extending previous results.

Figure 1

Comparison of $F$ (solid curve), $E_R=\text{co}F$ (partly dashed line, when convexification is necessary), and the numerical value of $E_R$ (dots) for the states ${\rho }_{3;0,1}\left(s\right)$, ${\rho }_{3;0,2}\left(s\right)$, and ${\rho }_{3;1,2}\left(s\right)$ (from top to bottom). Figures are reproduced from Ref. [23], where the conjecture was made.

Figure 2

Comparison of $F$ (solid), its convex hull, i.e., $E_R$ and the numerical value of $E_R$ for the states ${\rho }_{4;0,1}\left(s\right)$, ${\rho }_{4;1,2}\left(s\right)$, and ${\rho }_{4;1,3}\left(s\right)$ (from top to bottom). In these cases, $E_R=F$. Figures are reproduced from Ref. [23], where the conjecture was made.

Figure 3

Comparison of $F$ (solid), its convex hull, i.e., $E_R$ (partly dashed line, when convexification is necessary), and the numerical value of $E_R$ for the states ${\rho }_{4;0,2}\left(s\right)$ (top) and ${\rho }_{4;0,3}\left(s\right)$ (bottom). Figures are reproduced from Ref. [23], where the conjecture was made.

Figure 4

Comparison of $F$ (solid) and its convex hull, i.e., $E_R$ (dashed) for the state ${\rho }_{ab}\left(s\right)$.