On structural physical approximations and entanglement breaking maps

Very recently, a conjecture saying that the so-called structural physical approximations (SPAs) to optimal positive maps (optimal entanglement witnesses) give entanglement breaking (EB) maps (separable states) has been posed (Korbicz et al 2008 Phys. Rev. A 78 062105). The main purpose of this contribution is to explore this subject. First, we extend the set of entanglement witnesses supporting the conjecture. Then, we ask whether 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 Lambda, there exists an EB channel Phi such that the SPA of Lambda constructed with the aid of Phi is again an EB channel. Finally, we ask similar questions in the case of continuous variable systems. We provide a simple way of constructing SPA and prove that in the case of the transposition map it gives the EB channel.