An O(N) Algorithm of Separability for Two-Partite Arbitrarily Dimensional Pure States

How to discriminate entanglement state and separable state rapidly is a key task in quantum information theory. In this paper, we will give a simple separability criterion and a fast algorithm for bipartite pure state systems based on the two order minors of the coefficient matrices of quantum states. By our algorithm, it only needs at most O(d) times operations of multiplication and comparison to judge separability for two-partite pure states in a d dimensional Hilbert space. Furthermore, our algorithm can be easily generalized to multi-partite system. For n-partite pure states with dimension d, our algorithm only needs at most O(dln(d)) times operations of multiplication and comparison.