Experimental demonstration of optimal unambiguous two-out-of-four quantum state elimination

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.