  Great advances have been achieved in studying characteristics of entanglement
for fundamentals of quantum mechanics and quantum information processing.
However, even for N-qubit systems, the problem of entanglement criterion has
not been well solved. In this Letter, using the method of state decomposition
and high order singular value decomposition (HOSVD), we propose a necessary and
sufficient entanglement criterion for general N-qubit systems. As an example,
we apply our method to study the multi-qubit W state with white noise. We not
only obtain the separability critical point, which is tight and thus better
than known results, but also the separate states ensemble for decomposition.
More examples are presented to show our criterion is accurate, which is tighter
than the well-known positive partial transpose criterion. For two-qubit case,
we can provide an entanglement measurer which gives similar results with
concurrence up to a factor. Our results pave the way to solve the
entanglement-separability criterion for more general cases.
