  The fact that not all quantum observables are jointly measurable is one of
the major differences between quantum and classical theory. In the former,
non-commuting observables can only be simultaneously measured with limited
precision. We report on an experimental implementation of such a simultaneous
measurement of two non-commuting observables based on the framework of weak von
Neumann measurements. We use a photonic setup where the polarisation degree of
freedom acts as the system and the two components of the transversal position
correspond to the pointers of the measurement apparatuses. In addition, the
theory shows that these weak measurements demonstrate a counter-intuitive
non-monotonicity: weaker measurements can potentially reveal more information
than stronger ones.
