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 $O(d\ln(d))$ times operations of multiplication and comparison.