  The ability to post-select the outcomes of an experiment is a useful
theoretical concept and experimental tool. In the context of weak measurements
post-selection can lead to surprising results such as complex weak values
outside the range of eigenvalues. Usually post-selection is realized by a
projective measurement, which is hard to implement in ensemble systems such as
NMR. We demonstrate the first experiment of a weak measurement with
post-selection on an NMR quantum information processor. Our setup is used for
measuring complex weak values and weak values outside the range of eigenvalues.
The scheme for overcoming the problem of post-selection in an ensemble quantum
computer is general and can be applied to any circuit-based implementation.
This experiment paves the way for studying and exploiting post-selection and
weak measurements in systems where projective measurements are hard to realize
experimentally.
