Genuinely entangled symmetric states with no $N$-partite correlations

We investigate genuinely entangled $N$-qubit states with no $N$-partite correlations in the case of symmetric states. Using a tensor representation for mixed symmetric states, we obtain a simple characterization of the absence of $N$-partite correlations. We show that symmetric states with no $N$-partite correlations cannot exist for an even number of qubits. We fully identify the set of genuinely entangled symmetric states with no $N$-partite correlations in the case of three qubits, and in the case of rank-2 states. We present a general procedure to construct families for an arbitrary odd number of qubits.

Figure 1

Three-qubit symmetric states in the space of eigenvalues ${\alpha }_i$ of $A$. The orange plane (triangle down) corresponds to the condition $\mathrm{tr}A={\sum }_i{\alpha }_i=1$. Points inside the dashed blue circle defined by $\mathrm{tr}{A}^2={\sum }_i{\alpha }_i^2=1$ correspond to physical states $\rho \ge 0$. Points inside the green triangle up, defined by ${\alpha }_i\ge 0$, correspond to separable states. Points outside the green triangle up and inside the dashed blue circle correspond to genuinely entangled states. Points lying between the dashed circle and the solid red trilobe defined by ${max}_i{\alpha }_i={\sum }_i{\alpha }_i^2$ correspond to genuinely entangled states detected by the sufficient criterion (18).

Figure 2

Three-qubit (a) and five-qubit (b) symmetric states in the space of eigenvalues of $\rho$. The orange line ${\lambda }_0+{\lambda }_1=1/2$ for $N=3$, and the large orange triangle down for $N=5$ correspond to the condition $\mathrm{tr}\rho =2{\sum }_i{\lambda }_i=1$. Points where all ${\lambda }_i\ge 0$ correspond to physical states $\rho \ge 0$ and are depicted by the thick green line $0\le {\lambda }_0\le 1/2$ for $N=3$, the green triangle up for $N=5$. In (a) and (b), the equality in (50) is the dashed blue curve and the numerically solved criterion (18) is the solid red curve. In (a), the dotted straight line is the analytical result ${\lambda }_1=3{\lambda }_0$. In (b), only the intersection with the normalization plane is depicted.