QUROPE Quantum Information Processing and Communication in Europe

Very recently a conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking (EB) maps (separable states) has been posed [J. K. Korbicz et al., Phys. Rev. A 78, 062105 (2008)]. The main purpose of this contribution is to explore this subject. First, we extend the set of witnesses obeying the conjecture. Then, we ask if structural physical approximations constructed from other than the depolarizing channel maps also lead to some EB maps. We formulate and prove a weaker conjecture stating that for any positive map there exists an entanglement breaking map such that SPA constructed from it is entanglement breaking. Finally, we ask similar questions in the case of continuous variable systems. We provide a simple way of contraction of SPA, and prove that in the case of the transposition map it gives entanglement breaking channel.