Detection and typicality of bound entangled states

We derive an explicit analytic estimate for the entanglement of a large class of bipartite quantum states, which extends into bound entanglement regions. This is done by using an efficiently computable concurrence lower bound, which is further employed to numerically construct a volume of $3\times 3$ bound entangled states.

Figure 1

(Color online) (a) Parameter space of the “line” states, as defined in Eq. (9), for $\gamma =0$: The crosshatched dark gray area depicts states with positive partial transpose, while the quasipure lower bound ${\mathcal{C}}_{\text{qp}}$ [Eq. (5)] vanishes in the light gray area. The black area corresponds to states that have positive partial transpose and ${\mathcal{C}}_{\text{qp}}$ exhibit positive values and thus is associated with bound entangled states. The solid gray triangle encompasses the positivity region $\left(\rho >0\right)$. The curves refer to the borderlines in Eq. (10): the full line delimits the positive partial transposition region, and the dashed ellipses the bound entangled region, when over the PPT region. Note that some entangled states, with negative partial transposition, have ${\mathcal{C}}_{\text{qp}}=0$, but this was not observed in areas of bound entanglement. (b) Zoom into the bound entangled area shown in (a).

Figure 2

(Color online) Same as Fig. 1, for nonvanishing values of $\gamma$ in Eq. (9). Also in these cases a perfect agreement is observed between the areas detected by the lower bound and the exact areas defined by Eq. (10).

Figure 3

Parameter space of the “beyond line” states as defined in Eq. (11) for $\gamma =-1/12$. The color code is as before.

Figure 4

Distribution of quasi pure concurrence ${\mathcal{C}}_{\text{qp}}$ [Eq. (5)] and of the negativity $\mathcal{N}$ (inset) of $3\times 3$ random states, generated by admixture of Hilbert-Schmidt-distributed random mixed states ${\rho }_{\text{HS}}$ to the bound entangled reference “line” state ${\rho }_{\text{be}}$, according to Eq. (12). Saturation indicates the binpopulation of the histogram for a fixed $ϵ$.