  Symmetric informationally complete measurements are both important building
blocks in many quantum information protocols and the seminal example of a
generalised, non-orthogonal, quantum measurement. In higher-dimensional
systems, these measurements become both increasingly interesting and
increasingly complex to implement. Here, we demonstrate an integrated quantum
photonic platform to realize such a measurement on three-level quantum systems.
The device operates at the high fidelities necessary for verifying a genuine
many-outcome quantum measurement, performing near-optimal quantum state
discrimination, and beating the projective limit in quantum random number
generation. Moreover, it is programmable and can readily implement other
quantum measurements at similarly high quality. Our work paves the way for the
implementation of sophisticated higher-dimensional quantum measurements that go
beyond the traditional orthogonal projections.
