On Entanglement and Separability

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