We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in strong sense, definable P-points, tight MAD families selective independent families. As result, we obtain model which a=u=i=ℵ1<2ℵ0=ℵ2, each of a, u, i has Π11 witness there is Δ31 well-order the reals. Note both complexity witnesses above combinatorial cardinal characteristics, well are...

We establish a relation between concurrence and entanglement witnesses. In particular, we construct entanglement witnesses for three-qubit W and GHZ states in terms of concurrence and different set of operators that generate it. We also generalize our construction for multi-qubit states.