Characterizing multiparticle entanglement in symmetric N-qubit states via negativity of covariance matrices

We show that higher order intergroup covariances involving even number of qubits are necessarily positive semidefinite for N-qubit separable states, which are completely symmetric under permutations of the qubits. This identification leads to a family of sufficient conditions of inseparability based on the negativity of 2kth order intergroup covariance matrices (2k