  The question of whether quantum phenomena can be explained by classical
models with hidden variables is the subject of a long lasting debate. In 1964,
Bell showed that certain types of classical models cannot explain the quantum
mechanical predictions for specific states of distant particles. Along this
line, some types of hidden variable models have been experimentally ruled out.
An intuitive feature for classical models is non-contextuality: the property
that any measurement has a value which is independent of other compatible
measurements being carried out at the same time. However, the results of
Kochen, Specker, and Bell show that non-contextuality is in conflict with
quantum mechanics. The conflict resides in the structure of the theory and is
independent of the properties of special states. It has been debated whether
the Kochen-Specker theorem could be experimentally tested at all. Only
recently, first tests of quantum contextuality have been proposed and
undertaken with photons and neutrons. Yet these tests required the generation
of special quantum states and left various loopholes open. Here, using trapped
ions, we experimentally demonstrate a state-independent conflict with
non-contextuality. The experiment is not subject to the detection loophole and
we show that, despite imperfections and possible measurement disturbances, our
results cannot be explained in non-contextual terms.
