Measurement Uncertainty

Abstract

In Chap. 11 we have seen that noncommutativity prohibits the joint measurability of two sharp observables but not necessarily that of two unsharp observables. We have also observed that adding noise to two incompatible observables can turn them into jointly measurable observables. This leads to the idea of realising an approximate joint measurement of two incompatible sharp observables by means of a joint measurement of two compatible, possibly unsharp observables. In this chapter we develop tools to quantify the approximation errors and the degrees of unsharpness necessary to achieve compatible approximations. This will prepare the ground for formulating measurement uncertainty relations, which make precise Heisenberg’s intuitive formulations of 1927 (Heisenberg, Z Physik 43:172–198, 1927).