Universal methods for extending any entanglement witness from the bipartite to the multipartite case

Abstract

Any bipartite entanglement witness $W$ can be written as $W=c_{\sigma}I-\sigma$, where $\sigma$ is a quantum state, $I$ is the identity matrix, and $c_{\sigma}$ is a non-negative number. We present a general method to extend the given entanglement witness to multipartite cases via purification, partial purification, and direct tensor of the quantum state $\sigma$. Our methods extend $\sigma$ but leave the parameter $c_{\sigma}$ untouched. This is very valuable since the parameter is generally not easy to compute.