Quantum separable criterion based on characteristic function

TL;DR: A quantum separable criterion based on characteristic function is proposed. The criterion is constructed on the none-zero elements of the characteristic matrix after local filtering and rotation transforms. It is shown to be better than the realignment criterion for most of the states.

Abstract: Entanglement is the most important feature of quantum information science and it is closely related to the foundation of quantum mechanics. To determine a given bipartite quantum state to be entangled or not is by no means easy. We propose a separable criterion for bipartite quantum states based on their characteristic function or characteristic matrix. We use local filtering transform and local rotation transform to simplify the characteristic matrix, while keeping the positivity and entanglement property of the quantum state invariant. The development of Heisenberg-Weyl matrix basis makes the transforms possible. The local transforms greatly reduce the number of none-zero elements of characteristic matrix. The new criterion of separability then is constructed on the none-zero elements. We compare the entanglement detecting properties of our criterion with realignment criterion and modified correlation matrix criterion on $3\times~3$ bound entangled states. It is shown that our criterion is better than realignment criterion for most of the states.