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We present a tomographic method which requires only $4d-3$ measurementoutcomes to reconstruct \emph{any} pure quantum state of arbitrary dimension$d$. Using the proposed scheme we have experimentally reconstructed a largenumber 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\emph{a priori} assumption of purity can be certified within the same set ofmeasurements, what represents an improvement with respect to other similarmethods and contributes to answer the question of how many observables areneeded to uniquely determine any pure state.

A set of $4d-3$ observables to determine any pure qudit state