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

Abstract

Any bipartite entanglement witness $W$ can be written as $W=c_{\ddot{I}}ƒI-\ddot{I}ƒ$, where $\ddot{I}ƒ$ is a quantum state, $I$ is the identity matrix, and $c_{\ddot{I}}ƒ$ 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 $\ddot{I}ƒ$. Our methods extend $\ddot{I}ƒ$ but leave the parameter $c_{\ddot{I}}ƒ$ untouched. This is very valuable since the parameter is generally not easy to compute.

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