  In this paper, we derive a general formula of the tangle for pure states of
three qubits, and present three explicit local unitary (LU) polynomial
invariants. Our result goes beyond the classical work of tangle, 3-tangle and
von Neumann entropy of entanglement for Ac\'{\i}n et al.' Schmidt decomposition
(ASD) of three qubits by connecting the tangle, 3-tangle, and von Neumann
entropy for ASD with Ac\'{\i}n et al.'s LU invariants. In particular, our
result reveals a general relation among tangle, 3-tangle, and von Neumann
entropy, together with a relation among their averages. The relations can help
us find the entangled states satisfying distinct requirements for tangle,
3-tangle, and von Neumann entropy. Moreover, we obtain all the states of three
qubits of which tangles, concurrence, 3-tangle and von Neumann entropy don't
vanish and these states are endurable when one of three qubits is traced out.
We indicate that for the three-qubit W state, its average von Neumann entropy
is maximal only within the W SLOCC class, and that under ASD the three-qubit
GHZ state is the unique state of which the reduced density operator obtained by
tracing any two qubits has the maximal von Neumann entropy.
