  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.
