Entangled symmetric states ofNqubits with all positive partial transpositions

Abstract

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.

Nonlocality in many-body quantum systems detected with two-body correlators

“…By tuning the value of h, the ground state of (89) varies along all |D k n for every k. Observe that the interactions in (89) are over all possible pairs, so it is worth mentioning that Dicke states of n qubits can also be well approximated for low n as ground states of a one-dimensional ferromagnetic spin-1/2 chain with nearest-neighbor XXZ interactions [72]. The states |D k n are genuinely multipartite entangled for any 0 < k < n. Recall that any n-qubit symmetric state (pure or mixed) is entangled if, and only if, it is genuinely multipartite entangled [73,74]). These states are also generically genuinely multipartite nonlocal [75].…”

Separability of symmetric states and vandermonde decomposition

“…And we believe that the mixed separable state would have the same property, which is shown in the following lemmas 2 and 17. For symmetric states, we have the following result: Lemma 2 (Augusiak et al [45]). Suppose ρ is a multi-qudit symmetric state, then it is either genuine entangled or fully separable.…”

“…By corollary 5, ρ has exactly r product vectors in its range and this decomposition is unique. + If we apply the PPT criterion on ρ, according to the result in [45], we have the following result about the decomposition of multiqubit separable symmetric states.…”

“…Furthermore, Tóth et al [43,44] proved the existence of entangled PPT symmetric states in five-qubit and six-qubit symmetric system. Augusiak et al [45] characterized the PPT entangled states in the symmetric system consisting of an arbitrary number of qubits. In particular, the authors provide criteria for the separability of such states formulated in terms of their ranks.…”

Entanglement and the truncated moment problem

“…Various results on entanglement for symmetric states have been obtained in the literature [5][6][7][8][9]. For instance, criteria for certifying separability in symmetric mixed states of N qubits were found in [10].…”