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 $2k$th order intergroup covariance matrices ($2k\le N$) of symmetric $N$-qubit systems. These conditions have a simple structure and detect entanglement in all even partitions of the symmetric multiqubit system. The observables involved are feasible experimental quantities and do not demand full state determination through quantum state tomography.