  A core principle of quantum theory is that non-orthogonal quantum states
cannot be perfectly distinguished with single-shot measurements. However, it is
possible to exclude a subset of non-orthogonal states without error in certain
circumstances. Here we implement a quantum state elimination measurement which
unambiguously rules out two of four pure, non-orthogonal quantum states --
ideally without error and with unit success probability. This is a generalised
quantum measurement with six outcomes, where each outcome corresponds to
excluding a pair of states. Our experimental realisation uses single photons,
with information encoded in a four-dimensional state using optical path and
polarisation degrees of freedom. The prepared state is incorrectly ruled out up
to $3.3(2)\%$ of the time.
