Genuine multiparticle entanglement of permutationally invariant states

We consider the problem of characterizing genuine multiparticle entanglement for permutationally invariant states using the approach of positive partial transpose mixtures. We show that the evaluation of this necessary biseparability criterion scales polynomially with the number of particles. In practice, it can be evaluated easily up to ten qubits and improves existing criteria significantly. Finally, we show that our approach solves the problem of characterizing genuine multiparticle entanglement for permutationally invariant three-qubit states.

Figure 1

Here we see, for three particle states, a scheme of the biseparable states with respect to the three possible bipartitions, $A|BC$, $B|AC$, and $C|AB$ (surrounded by solid blue lines). Each of these is contained in the PPT states for each respective bipartition (surrounded by dashed green lines). The larger solid blue line enveloping the other solid blue lines represents the set of biseparable states, and similarly, the larger dashed green line defines the set of PPT mixtures. This scheme can be generalized for higher numbers of particles.

Figure 2

Due to the form given by Eq. (14), a PI state has a block diagonal structure in a suitably ordered basis. Each block $B_j$ appears $\dim \left({\mathcal{K}}_j\right)$ times in the diagonal.

Figure 3

Comparison between the PPT mixture criterion (solid symbols) and the criterion of Huber et al. [13] (open symbols) for the white-noise tolerance for Dicke states $|D_{N,k}\rangle$. We show the white-noise tolerances for $N$ up to ten qubits and $k$ up to $N/2$. The PPT mixture criterion is more robust to white noise, and in some cases, the difference of the white-noise tolerance between both criteria is very significant, reaching values larger than 40$\%$.

Figure 4

Here, we show which three-qubit states (lower triangular) as defined by Eq. (42) are detected by the PPT mixture criterion as genuinely multipartite entangled (area outside the solid blue line). We compare it to the criteria presented in Refs. [12, 14], which consist of two inequalities, one optimized for the GHZ state and the other for the $W$ state (dashed red lines). The PPT mixture criterion represents a significant improvement and is optimal for three-qubit PI states, so the states inside the solid line are biseparable.

Figure 5

The same as Fig. 4, but for eight-qubit states. The PPT mixture criterion represents a very significant improvement, but note that here we cannot prove that the PPT mixture criterion is necessary and sufficient for biseparability.