We propose two necessary sufficient (NS) criteria to decide the separability
of quantum states. They follow from two independent ideas: i) the
Bloch-sphere-like-representation of states and ii) the proportionality of lines
(rows, columns etc.) of certain multimatrix [1] associated with states. The
second criterion proposes a natural way to determine the possible partial (or
total, when possible) factorization of given multipartite state and in a sense
can be used to determine the structure of the entanglement. We also introduce
three entanglement measures based on the proposed new characterizations of
entanglement. We then discuss the second criterion mentioned above in the
language of density matrix which is an inevitable language especially for mixed
states. We develop factorization algorithm for quantum states comprising a
useful technique to express any given quantum state as a product of maximally
entangled factors. Finally, we discuss algorithm in the last section which is
useful to speed-up the factorization process.Comment: 27 pages. Revised Section