  Several families of states such as Werner states, Bell-diagonal states and
Dicke states are useful to understand multipartite entanglement. Here we
present a [2^(N+1)-1]-parameter family of N-qubit "X states" that embrace all
those families, generalizing previously defined states for two qubits. We also
present the algebra of the operators that characterize the states and an
iterative construction for this algebra, a sub-algebra of su(2^(N)). We show
how a variety of entanglement witnesses can detect entanglement in such states.
Connections are also made to structures in projective geometry.
