Comprehensive analysis of quantum pure-state estimation for two-level systems

Given $N$ identical copies of the state of a quantum two-level system, we analyze its optimal estimation. We consider two situations: general pure states and (pure) states restricted to lie on the equator of the Bloch sphere. We perform a complete and comprehensive analysis of the optimal schemes based on local measurements, and give results (optimal measurements, maximum fidelity, etc.) for arbitrary $N$, not necessarily large, within the Bayesian framework. We also make a comparative analysis of the asymptotic limit of these results with those derived from a (pointwise) Cramér-Rao type of approach. We give explicit schemes based on local measurements and no classical communication that saturate the fidelity bounds of the most general collective schemes.

Figure 1

Average fidelities in terms of the number of copies in the 2D case for the optimal collective measurement (solid line), tomographic OG (dot-dashed line), tomographic FG (dashed line), and a simulation of the greedy scheme (dots).

Figure 2

Scaled error ${ϵ}_N=N(1-F)$ in the 2D case for collective (solid line), OG (dot-dashed line), FG (dashed line), and greedy (diamonds) schemes.

Figure 3

Scaled error ${ϵ}_N=N(1-F)$ in the 3D case for collective (solid line), tomographic OG (dot-dashed line), tomographic FG (dashed line), and greedy (diamonds) schemes.

Figure 4

Scaled error ${ϵ}_N$ of the RD scheme (triangles) as compared to the OG (dot-dashed line) and the optimal collective scheme (solid line).