Qudit state estimation with a fixed set of bases

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