Constructing all entanglement witnesses from density matrices

Abstract

We demonstrate a general procedure to construct entanglement witnesses for any entangled state. This procedure is based on the trace inequality and a general form of entanglement witnesses, which is in the form $W=\rho-c_{\rho} I$, where $\rho$ is a density matrix, $c_{\rho}$ is a non-negative number related to $\rho$, and $I$ is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio{\l}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=\rho-c_{\rho}I$ in operation, as well as in theory.