  We investigate the norms of the Bloch vectors for any quantum state with
subsystems less than or equal to four. Tight upper bounds of the norms are
obtained, which can be used to derive tight upper bounds for entanglement
measure defined by the norms of Bloch vectors. By using these bounds a
trade-off relation of the norms of Bloch vectors is discussed. Theses upper
bounds are then applied on separability. Necessary conditions are presented for
different kinds of separable states in four-partite quantum systems. We further
present a complete classification of quantum states for four qudits quantum
systems.
