Collective multipolelike signatures of entanglement in symmetric $N$-qubit systems

A cogent theory of collective multipolelike quantum correlations in symmetric multiqubit states is presented by employing SO(3) irreducible spherical tensor representation. An arbitrary bipartite division of this system leads to a family of inequalities to detect entanglement involving averages of these tensors expressed in terms of the total system angular momentum operator. Implications of this theory to the quantum nature of multipolelike correlations of all orders in the Dicke states are deduced. A selected set of examples illustrate these collective tests. Such tests detect entanglement in macroscopic atomic ensembles, where individual atoms are not accessible.

Figure 1

Threshold value $x_{\text{min}}$, evaluated from the multipole inseparability condition $C^{\left(2\kappa \right)}\left(\kappa ,\kappa \right)<0$, with $\kappa =1$ to $\kappa =5$ (uppermost curve to the lower ones, respectively) for the states ${\widehat{\rho }}_1$ [see (a)] and ${\widehat{\rho }}_2$ [see (b)], as a function of the number of atoms $N$. (c) $x_{\text{min}}$ obtained from the negativity of the highest order covariance matrix $C^{\left(N\right)}\left(N∕2,N∕2\right)$ for the states ${\widehat{\rho }}_1$ $(+)$, ${\widehat{\rho }}_2$ $(◻)$, and ${\widehat{\rho }}_3$ $(\times )$.