Coexistence of quantum effects

The witness set W(A, B) for two quantum effects A, B is the set of effects that verify the coexistence of A and B. We first study the properties of W(A, B). We then introduce a measure of sharpness Sn(A) for an n-dimensional quantum effect A. We show that S2(A) has natural properties that a measure of sharpness should possess and conjecture that Sn(A) also has these properties for n > 2. Finally, we employ S2(A) to obtain a characterization for the coexistence of two qubit effects in terms of operational parameters.