Criteria for entanglement and separability of discrete quantum states

Entanglement is an important quantum resource, which can be used in quantum teleportation and quantum computation. How to judge and measure entanglement or separability has become a basic problem in quantum information theory. In this paper, by analyzing the properties of generalized ring $\mathbb{Z}[i]^{{2}^{n}}$, a new method is presented to judge the entanglement or separability of any quantum state in the discrete quantum computing model proposed by Gatti and Lacalle. Different from previous criteria based on matrices, it is relatively simple to operate in mathematical calculation. And if a quantum state is separable, it can calculate the separable mathematical expression. Taking $n=2,3$ as examples, the concrete forms of all separable states in the model are presented. It provides a new research perspective for the discrete quantum computing model.