For multipartite states, we consider a notion of
D symmetry. For a system of
N qubits, it coincides with the usual permutational symmetry. In the case of
N qudits ($d\ensuremath{\ge}3$), the
D symmetry is stronger than the permutational one. For the space of all
D-symmetric vectors in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, we define a basis composed of vectors ${|{R}_{N,d;k}\ensuremath{\rangle}:\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}k\ensuremath{\le}N(d\ensuremath{-}1)}$ which are analogs of Dicke states. The aim of this paper is to discuss the problem of separability of
D-symmetric states which are diagonal in the basis ${|{R}_{N,d;k}\ensuremath{\rangle}}$. We show that if
N is even and $d\ensuremath{\ge}2$ is arbitrary then a positive partial transposition property is a necessary and sufficient condition of separability for
D-invariant diagonal states. In this way, we generalize results obtained by Yu [Phys. Rev. A 94, 060101(R) (2016)] and Wolfe and Yelin [Phys. Rev. Lett. 112, 140402 (2014)]. Our strategy is to use some classical mathematical results on a moment problem.
The structural study of entanglement in multipartite systems is hindered by the lack of necessary and sufficient operational criteria able to discriminate among the various entanglement properties of a given mixed state. Here, we pursue a different route to the study of multipartite entanglement based on the closeness of a multipartite state to the set of separable ones. In particular, we analyze multipartite diagonal symmetric
N-qubit states and provide the analytical expression for their best separable approximation (BSA, [Phys. Rev. Lett. 80, 2261 (1998)]), that is, their unique convex decomposition into a separable part and an entangled part, with maximal weight of the separable part.
High-dimensional quantum entanglement and the advancements in their experimental realization provide a playground for fundamental research and eventually lead to quantum technological developments. The Horodecki criterion determines whether a state violates the Clauser-Horne-Shimony-Holt (CHSH) inequality for a two-qubit entangled state, solely from the state parameters. However, it remains a challenging task to formulate similar necessary and sufficient criteria for a high-dimensional entangled state for the violation of a suitable Bell inequality. Here we develop a Horodecki-like criterion based on the state parameters of arbitrary two-qudit states to violate a two-outcome Bell inequality involving ${2}^{n\ensuremath{-}1}$ and
n measurement settings for Alice and Bob, respectively. This inequality reduces to the well-known CHSH and Gisin elegant Bell inequalities for
n=2 and 3, respectively. While the proposed criterion is sufficient to violate the Bell inequality, it becomes necessary as well for the following cases: (i)
m copies of Bell-diagonal states for arbitrary
n, (ii) nondecomposable states whose correlation matrix is diagonalized by local unitaries, and (iii) for any arbitrary two-qubit state when
n=3, where the maximal value of the Bell functional is achieved with Bob's measurements being pairwise anticommuting. For any states, we derive the constraints on Alice's measurements in achieving the maximum quantum violation for this inequality.
Following the linear programming prescription of Jafarizadeh et al. [Phys. Rev. A 72, 062106 (2005)], the $d\ensuremath{\bigotimes}d$ Bell diagonal entanglement witnesses are provided. By using Jamiolkowski isomorphism, it is shown that the corresponding positive maps are the generalized qudit Choi maps. Also by manipulating particular $d\ensuremath{\bigotimes}d$ Bell diagonal separable states and constructing corresponding bound entangled states, it is shown that the obtained $d\ensuremath{\bigotimes}d$ Bell states diagonal entanglement witnesses (consequently qudit Choi maps) are nondecomposable in a certain range of their parameters.
The structural study of entanglement in multipartite systems is hindered by the lack of necessary and sufficient operational criteria able to discriminate among the various entanglement properties of a given mixed state. Here, we pursue a different route to the study of multipartite entanglement based on the closeness of a multipartite state to the set of separable ones. In particular, we analyze multipartite diagonal symmetric N qubit states and provide the analytical expression for their Best Separable Approximation (BSA [Phys. Rev. Lett. 80, 2261 (1998)]), that is, their unique convex decomposition into a separable part and an entangled one with maximal weight of the separable one.