Sequential measurements of non-commuting observables with quantum controlled interactions

The origin of non-classical correlations is difficult to identify since the uncertainty principle requires that information obtained about one observable invariably results in the disturbance of any other non-commuting observable. Here, this problem is addressed by investigating the uncertainty trade-off between measurement errors and disturbance for measurement interactions controlled by the state of a single qubit, where the measurement is described by a quantum coherent superposition of a fully projective measurement and the identity operation. It is shown that the measurement statistics obtained from a quantum controlled measurement of A followed by a projective measurement of B can be explained in terms of a simple combination of resolution and back-action errors acting on an intrinsic joint probability of the non-commuting observables defined by the input state of the system. These intrinsic joint probabilities are consistent with the complex-valued joint probabilities recently observed in weak measurements of quantum systems and provide direct evidence of non-commutativity in the form of imaginary correlations between the non-commuting operators. In quantum controlled measurements, these imaginary correlations can be converted into well-defined contributions to the real measurement statistics, allowing a direct experimental observation of the less intuitive aspects of quantum theory.