Measuring measurement–disturbance relationships with weak values

Using formal definitions for the measurement precision is an element of and the disturbance (measurement back-action) eta, Ozawa (2003 Phys. Rev. A 67 042105) has shown that Heisenberg's claimed relation between these quantities is false in general. Here, we show that the quantities introduced by Ozawa can be determined experimentally, using no prior knowledge of the measurement under investigation-both quantities correspond to the root-meansquared difference given by a weak-valued probability distribution. We propose a simple three-qubit experiment that can illustrate the failure of Heisenberg's measurement-disturbance relation and the validity of an alternative relation proposed by Ozawa.