The witness set of coexistence of quantum effects and its preservers

One of unsolved problems in quantum measurement theory is to characterize coexistence of quantum effects. In this paper, applying positive operator matrix theory, we give a mathematical characterization of the witness set of coexistence of quantum effects and obtain a series of properties of coexistence. We also devote to characterizing bijective morphisms on quantum effects leaving the witness set invariant. Furthermore, applying linear maps preserving commutativity, which are characterized by Choi, Jafarian and Radjavi [Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987), 227--241.], we classify multiplicative general morphisms leaving the witness set invariant on finite dimensional Hilbert space effect algebras.