Disturbance in weak measurements and the difference between quantum and classical weak values

The role of measurement-induced disturbance in weak measurements is of central importance for the interpretation of the weak value. Uncontrolled disturbance can interfere with the postselection process and make the weak value dependent on the details of the measurement process. Here we develop the concept of a generalized weak measurement for classical and quantum mechanics. The two cases appear remarkably similar, but we point out some important differences. A priori it is not clear what the correct notion of disturbance should be in the context of weak measurements. We consider three different notions and get three different results: (1) For a "strong" definition of disturbance, we find that weak measurements are disturbing. (2) For a weaker definition we find that a general class of weak measurements is nondisturbing, but that one gets weak values which depend on the measurement process. (3) Finally, with respect to an operational definition of the "degree of disturbance," we find that the AAV weak measurements are the least disturbing, but that the disturbance is always nonzero.