  We present a complete set of local unitary invariants for generic multi-qubit
systems which gives necessary and sufficient conditions for two states being
local unitary equivalent. These invariants are canonical polynomial functions
in terms of the generalized Bloch representation of the quantum states. In
particular, we prove that there are at most 12 polynomial local unitary
invariants for two-qubit states and at most 90 polynomials for three-qubit
states. Comparison with Makhlin's 18 local unitary invariants is given for
two-quibit systems.
