Eur. Phys. J. D 37, 141-150 (2006)
https://doi.org/10.1140/epjd/e2005-00271-9

Qudit state estimation with a fixed set of bases

Abteilung Quantenphysik, University Ulm, 89069 Ulm, Germany

Corresponding author: ^a Thorsten.Bschorr@uni-ulm.de

Received: 14 July 2005
Published online: 4 October 2005

Abstract

We analyse the estimation of a pure d-dimensional quantum state with a finite number of measurements and compare several estimation schemes. In this paper we concentrate on consecutive von Neumann measurements on a finite number of identically prepared systems in dimensions d=2, d=4 and d=8. We propose two schemes with different types of fixed measurement directions. Inspired by integration theory our first approach uses the Halton sequence (a so-called quasi-Monte Carlo sequence) to obtain measurement directions (`sampling points') with high uniformity over the configuration space. Our second approach extends this idea and optimises the distribution of the measurement directions to yield a rather high fidelity in quantum state estimation. This optimisation results in a uniform distribution of the directions and large quantum distances between the directions. Furthermore we establish a link to mutually unbiased bases.