Stable pure state quantum tomography from five orthonormal bases

¹ DIME, Università di Genova - Via Magliotto 2, I-17100 Savona, Italy
² Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku FI-20014 Turku, Finland
³ Department of Mathematics, Technische Universität München - D-85748 Garching, Germany
⁴ Dipartimento di Matematica, Politecnico di Milano - Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
⁵ I.N.F.N., Sezione di Milano - Via Celoria 16, I-20133 Milano, Italy

Received: 25 May 2016
Accepted: 6 August 2016

Abstract

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. (Phys. Rev. Lett., 115 (2015) 090401). 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.