  For any finite dimensional Hilbert space, we construct explicitly five
orthonormal bases such that the corresponding measurements allow for efficient
tomography of an arbitrary pure quantum state. This means that such
measurements can be used to distinguish an arbitrary pure state from any other
state, pure or mixed, and the pure state can be reconstructed from the outcome
distribution in a feasible way. The set of measurements we construct is
independent of the unknown state, and therefore our results provide a fixed
scheme for pure state tomography, as opposed to the adaptive (state dependent)
scheme proposed by Goyeneche et al. in [Phys. Rev. Lett. 115, 090401 (2015)].
We show that our scheme is robust with respect to noise in the sense that any
measurement scheme which approximates these measurements well enough is equally
suitable for pure state tomography. Finally, we present two convex programs
which can be used to reconstruct the unknown pure state from the measurement
outcome distributions.
