On structural physical approximations and entanglement breaking maps

Very recently a conjecture saying that the so-called structural physical approximations (SPAa) to optimal positive maps (optimal entanglement witnesses) give entanglement breaking (EB) maps (separable states) has been posed [J. K. Korbicz {\it et al.}, Phys. Rev. A {\bf 78}, 062105 (2008)]. The main purpose of this contribution is to explore this subject. First, we extend the set of entanglement witnesses (EWs) supporting the conjecture. Then, we ask if SPAs constructed from other than the depolarizing channel maps also lead to EB maps and show that in general this is not the case. On the other hand, we prove an interesting fact that for any positive map $Λ$ there exists an EB channel $Φ$ such that the SPA of $Λ$ constructed with the aid of $Φ$ is again an EB channel. Finally, we ask similar questions in the case of continuous variable systems. We provide a simple way of construction of SPA and prove that in the case of the transposition map it gives EB channel.

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