Entanglement and nonlocality in diagonal symmetric states of $N$ qubits

We analyze entanglement and nonlocal properties of the convex set of symmetric $N$-qubit states which are diagonal in the Dicke basis. First, we demonstrate that within this set, semidefinite positivity of partial transposition (PPT) is necessary and sufficient for separability—which has also been reported recently by Yu [Phys. Rev. A 94, 060101(R) (2016)]. Furthermore, we show which states among the entangled diagonal symmetric are nonlocal under two-body Bell inequalities. The diagonal symmetric convex set contains a simple and extended family of states that violate the weak Peres conjecture, being PPT with respect to one partition but violating a Bell inequality in such partition. Our method opens directions to address entanglement and nonlocality on higher dimensional symmetric states, where presently very few results are available.

Figure 1

Representation of the PPT-DS region for $N=4$. The picture corresponds to $s=p_0+p_4=0.6$. Notice the shape of the region of separable states is independent of the value of $s$.

Figure 2

The generic PPT region given by Eqs. (12). Generic separable states are of rank $(5,8,9)$ (dark area), while extremal separable states are of rank $(5,6,6),(5,8,8)$, and $(5,7,7)$. Point A is an edge entangled state that satisfies ${\text{PPT}}_{{\mathrm{\Gamma }}_1}>0$ but ${\text{PPT}}_{{\mathrm{\Gamma }}_2}<0$.

Figure 3

Device angles for which there is Bell violation of Eq. (14) for the Dicke states $\left|D_{1,3}^4\right.〉$ (dark) and $\left|D_2^4\right.〉$ (dark + light). The device angle that corresponds to the maximal Bell violation is indicated.

Figure 4

Nonlocality of DS states for $p_0=p_4=0.1$. Gray areas correspond to ${\rho }_{\text{DS}}$ which are nonlocal, while white areas depict local states. PPT denotes separable states. By ${\text{PPT}}_{1|3}$ we refer to ${\rho }_{\text{DS}}$ such that it is PPT with respect to partition $1|3$ but NPT with respect to partition $2|2$. The boundaries of the PPT region obtained by numerically evaluating Bell inequalities are in one-to-one correspondence with the regions bounded by Eqs. (12).