  Based on weak measurement technology, we propose the first three-observer
dimension witness protocol in a prepare-and-measure setup. By applying the
dimension witness inequality based on the quantum random access code and the
nonlinear determinant value, we demonstrate that double classical dimension
witness violation is achievable if we choose appropriate weak measurement
parameters. Analysis of the results will shed new light on the interplay
between the multi-observer quantum dimension witness and the weak measurement
technology, which can also be applied in the generation of
semi-device-independent quantum random numbers and quantum key distribution
protocols.
