The structural physical approximations and optimal entanglement witnesses

We introduce the notions of positive and copositive types for entanglement witnesses, depending on the distance to the positive part and copositive part. An entanglement witness $W$ is of positive type if and only if its partial transpose $W^Γ$ is of copositive type. We show that if the structural physical approximation of $W$ is separable then $W$ should be of copositive type, and the SPA of $W^Γ$ is never separable unless $W$ is of both positive and copositive type. This shows that the SPA conjecture is meaningful only for those of copositive type. We provide examples to show that the SPA conjecture fails even for the case of copositive types.

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