Constructing all entanglement witnesses from density matrices

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 $W=ρ-c_ρ I$, where $ρ$ is a density matrix, $c_ρ$ is a non-negative number related to $ρ$, and $I$ is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamiołkowski 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 $W=ρ-c_ρI$ in operation, as well as in theory.

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