Entanglement monotones and transformations of symmetric bipartite states

The primary goal in the study of entanglement as a resource theory is to find conditions that determine when one quantum state can or cannot be transformed into another via local operations and classical communication operations. This is typically done through entanglement monotones or conversion witnesses. Such quantities cannot be computed for arbitrary quantum states in general, but it is useful to consider classes of symmetric states for which closed-form expressions can be found. In this paper, we show how to compute the convex roof of any entanglement monotone for all Werner states. The convex roofs of the well-known Vidal monotones are computed for all isotropic states, and we show how this method can generalize to other entanglement measures and other types of symmetries as well. We also present necessary and sufficient conditions that determine when a pure bipartite state can be deterministically converted into a Werner state or an isotropic state.

Figure 1

Schematic of the $OO$-invariant states ${\rho }_O(a,b)$, as defined in Eq. (9). The shaded region represents the separable states. The one-dimensional subfamilies of Werner and isotropic states are also shown. Convex roof entanglement monotones can be computed for states in regions A and B, as discussed in Sec. 3d. It remains unknown how to compute convex roofs on states in region C for an arbitrary entanglement monotone.

Figure 2

Schematic of the phase-permutation Werner states. The separable region is shown in gray. The one-dimensional family of states with $b=\frac{d-1}{d+1}(1-a)$ is made up of the well-known Werner states. As shown in Sec. 3d, the convex roof of any entanglement monotone can be computed for any state in region A. It remains unknown how to compute convex roofs on states in region B for an arbitrary entanglement monotone.

Figure 3

Schematic of the phase-permutation isotropic states. The separable region is shown in gray. The one-dimensional family of states with $b=1-(d+1)a$ is made up of the isotropic states. As shown in Sec. 3c, the convex roof of any entanglement monotone can be computed for any state in region B. It remains unknown how to compute convex roofs on states in region A for an arbitrary entanglement monotone.

Figure 4

The convex roof of the Vidal monotones $E_1, E_2, E_3$, and $E_4$ evaluated on isotropic states with dimension $d=5$.

Figure 5

Example values of $t$ from Eq. (31) (solid line) and Eq. (32) (dashed line) as functions of $b$ for $d=5$.

Figure 6

Convex roofs of the Vidal monotones $E_k$ (for $d=5$ and $k=1,2,3,4$) evaluated on regions A and B of the $OO$-invariant states. For the surfaces on the left-hand side, $k$ varies from 1 to 4 from the upper to the lower level. Only $E_1$ is nonvanishing on the right-hand side.

Figure 7

An example of the witness in Eq. (39) computed for $d=3$ and $\mathbf{\lambda }=(\frac{6}{10},\frac{3}{10},\frac{1}{10})$. It appears that $W(\mathbf{\lambda },b)<0$ whenever $b>0.895$.