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Entangled symmetric states of N qubits with all positive partial transpositions

From both theoretical and experimental points of view symmetric states constitute an important class of multipartite states. Still, entanglement properties of these states, in particular those with positive partial transposition (PPT), lack a systematic study. Aiming at filling in this gap, we have recently affirmatively answered the open question of existence of four-qubit entangled symmetric states with positive partial transposition and thoroughly characterized entanglement properties of such states [J. Tura et al., Phys. Rev. A 85, 060302(R) (2012)] With the present contribution we continue on characterizing PPT entangled symmetric states. On the one hand, we present all the results of our previous work in a detailed way. On the other hand, we generalize them to systems consisting of arbitrary number of qubits. In particular, we provide criteria for separability of such states formulated in terms of their ranks. Interestingly, for most of the cases, the symmetric states are either separable or typically separable. Then, edge states in these systems are studied, showing in particular that to characterize generic PPT entangled states with four and five qubits, it is enough to study only those that assume few (respectively, two and three) specific configurations of ranks. Finally, we numerically search for extremal PPT entangled states in such systems consisting of up to 23 qubits. One can clearly notice regularity behind the ranks of such extremal states, and, in particular, for systems composed of odd number of qubits we find a single configuration of ranks for which there are extremal states.