Set of 4d-3 observables to determine any pure qudit state.

We present a tomographic method which requires only 4d-3 measurement outcomes to reconstruct any pure quantum state of arbitrary dimension d. Using the proposed scheme, we have experimentally reconstructed a large number of pure states of dimension d=7, obtaining a mean fidelity of 0.94. Moreover, we performed numerical simulations of the reconstruction process, verifying the feasibility of the method for higher dimensions. In addition, the a priori assumption of purity can be certified within the same set of measurements, which represents an improvement with respect to other similar methods and contributes to answering the question of how many observables are needed to uniquely determine any pure state.