Non-disturbance for fuzzy quantum measurements

We consider three criteria for describing non-disturbance between quantum measurements. While previous discussions of these criteria considered sharp measurements, we shall treat more general measurements that may be unsharp (or fuzzy). It has been shown that in the sharp case, the three criteria are equivalent to compatibility of the measurements. We shall show that only the third criterion is equivalent to compatibility for the general fuzzy case. Moreover, the first two criteria are not symmetric relative to the two measurements.