Constructing optimal entanglement witnesses

We provide a class of indecomposable entanglement witnesses. In 4 x 4 case, it reproduces the well-known Breuer-Hall witness. We prove that these witnesses are optimal and atomic, i.e., they are able to detect the weakest quantum entanglement encoded into states with positive partial transposition. Equivalently, we provide a construction of indecomposable atomic maps in the algebra of 2k x 2k complex matrices. It is shown that their structural physical approximations give rise to entanglement breaking channels. This result supports recent conjecture by Korbicz et al. [Phys. Rev. A 78, 062105 (2008)].