Quantum state estimation with informationally overcomplete measurements

We study informationally overcomplete measurements for quantum state estimation so as to clarify their tomographic significance as compared with minimal informationally complete measurements. We show that informationally overcomplete measurements can improve the tomographic efficiency significantly over minimal measurements when the states of interest have high purities. Nevertheless, the efficiency is still too limited to be satisfactory with respect to figures of merit based on monotone Riemannian metrics, such as the Bures metric and quantum Chernoff metric. In this way, we also pinpoint the limitation of nonadaptive measurements and motivate the study of more sophisticated measurement schemes. In the course of our study, we introduce the best linear unbiased estimator and show that it is equally efficient as the maximum likelihood estimator in the large sample limit. This estimator may significantly outperform the canonical linear estimator for states with high purities. It is expected to play an important role in experimental designs and adaptive quantum state tomography besides its significance to the current study.

Figure 1

Tomographic efficiencies of the CLE, BLUE, and MLE. The scaled MSEs of these estimators are determined by numerical simulation of the cube measurement (cf. Sec. 4) on a qubit state with random Bloch vector $\mathit{s}=(0.6886,0.1137,-0.5025)$. Each data point is an average over 1000 repetitions. BLUE1 assumes the knowledge of the true state in computing the reconstruction operators, while BLUE2 uses frequencies instead of probabilities in relevant formulas. The theoretical scaled MSEs of the CLE and BLUE are shown as dashed line and solid line, respectively. Also plotted are pairwise scaled MSEs (multiplied by a factor of 10 for ease of viewing) among BLUE1, BLUE2, and MLE. The figure indicates that the three estimators are almost equally efficient as long as $N$ is not too small.

Figure 2

Tomographic efficiencies of covariant measurements. The true states have the form in Eq. (44) with $r=1$ and $d=2,3,...,6$ (from bottom to top). The scaled MSB diverges in the limit $s\to 1$, in which case the states are rank deficient. For comparison, the dashed lines show the performances of covariant measurements under canonical linear reconstruction. In plot (a), they also represent the performances of the optimal minimal IC measurements (that is SIC measurements) with respect to the scaled MSE.

Figure 3

Uncertainty ellipses of the canonical reconstruction and the optimal reconstruction. The uncertainty ellipses are associated with the marginal distributions on the $x\text{−}z$ plane of the Bloch ball resulting from mutually unbiased measurements on a family of states, each repeated 300 times. The optimal reconstruction reduces the sizes of the uncertainty ellipses at the prize of losing the covariance property.

Figure 4

Tomographic efficiencies of the SIC, MUB, cube, and covariant measurements in qubit state estimation. (a) Average scaled MSE; (b) average scaled MSB; (c) average log volume (with respect to the HS metric) of the scaled uncertainty ellipsoid. The average scaled MSBs diverge in the pure-state limit for all four measurements. For the covariant measurement, the log volume diverges to $-\infty$ (that is, the volume vanishes) in the pure-state limit. As comparison, the curve “Iso” shows the common performance of isotropic measurements (including MUB, cube, and covariant measurements) under canonical linear reconstruction. It coincides with the curve “SIC” in plots (a) and (b) since under canonical linear reconstruction isotropic measurements are as efficient as the SIC measurement in qubit state estimation with respect to the average scaled MSE and MSB.