Stable pure state quantum tomography from five orthonormal bases

Abstract

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