We investigate a class of partially device-independent quantum key distribution protocols based on a prepare-and-measure setup which simplifies their implementation. The security of the protocols is based on the assumption that Alice's prepared states have limited overlaps, but no explicit bound on the Hilbert space dimension is required. The protocols are therefore immune to attacks on Bob's device, such as blinding attacks. The users can establish a secret key while continuously monitoring the correct functioning of their devices through observed statistics. We report a proof-of-principle demonstration, involving mostly off-the-shelf equipment, as well as a high-efficiency superconducting nanowire detector. A positive key rate is demonstrated over a 4.8 km low-loss optical fiber with finite-key analysis. The prospects of implementing these protocols over longer distances is discussed.
