Choice of measurement sets in qubit tomography

Optimal generalized measurements for state estimation are well understood. However, practical quantum state tomography is typically performed using a fixed set of projective measurements, and the question of how to choose these measurements has been largely unexplored in the literature. In this work, we develop theoretical asymptotic bounds for the average fidelity of pure qubit tomography using measurement sets whose axes correspond to faces of Platonic solids. We also present comprehensive simulations of maximum likelihood tomography for mixed qubit states using the Platonic solid measurements. We show that overcomplete measurement sets can be used to improve the accuracy of tomographic reconstructions.

Figure 1

(Color online) Performance of tomographically reconstructed states on the Bloch sphere for two measurement sets. The top set is a popular set of measurements (James4) used in the literature [15], the bottom set is a more isotropic set composed of measurements along the three spatial axes (a cube measurement set); see text for details. For each target state, the average fidelity with 400 reconstructed states is plotted by color. The ensemble of reconstructed states was generated by adding Possonian noise to a simulated experiment with an average total of 4000 counts, and then performing maximum-likelihood tomography.

Figure 2

(Color online) Plot of probability density against Bloch radius for the Bures and Chernoff priors over states. Both distributions have densities that increase with state purity. The Chernoff prior is slightly more skewed toward the pure states than the Bures prior.

Figure 3

(Color online) Numerical simulations of atomic point fidelities for states rotated $45\text{degrees}$ about the $x$ axis from the positive $z$ axis. Each graph corresponds to a different radius on the Bloch sphere. The vertical axis is the number of copies of the state used in the tomography. The horizontal axis is one minus the average point fidelity. The cube measurement set was used for these simulations. Observe mixed states always have a $1∕N$ scaling. States that are close to pure start (after we get to a sufficiently large $N$ to obtain a sensible state estimate) with a $1∕\sqrt{N}$ dependence and transition to $1∕N$ scaling with larger $N$. This knee occurs at larger $N$ for purer states. The statistical theory of Eq. (15) is plotted for $r=0.5$.

Figure 4

The single photonic qubit experimental configuration consists of a quarter-wave plate and a half-wave plate to set the basis, and a polarizing beam splitter followed by a photon detector to analyze in that basis. We call this a single-detector configuration. An additional detector can be placed on the reflected port in a dual-detector configuration. This will produce the same count statistics as in the atomic case after waiting for a fixed number of counts for each basis setting.

Figure 5

(Color online) Tomography simulation results with a Bures prior. In (a) and (b), the number of copies of the state are plotted against $1-F_{\mathrm{av}}$, where $F_{\mathrm{av}}$ is the average fidelity. To read, select the desired average fidelity on the horizontal axis, and trace vertically to find the number of copies of the state required to achieve that fidelity. Similarly, (c) and (d) plot the photon flux $N_f$ multiplied by the total time $t$ against $1-F_{\mathrm{av}}$. N.B. the total time $t$ is equally divided between the different measurement settings. Each plot shows the tomography performance of various Platonic solids. The individual figures are tomography for (a) single atomic qubit states (the tetrahedron cannot be measured in this configuration), (b) two atomic qubit states, (c) single photonic qubit with single-detector configuration, and (d) two photonic qubit with single-detector configuration.

Figure 6

(Color online) Tomography simulation results with the prior induced by the quantum Chernoff bound. In (a) and (b), the number of copies of the quantum state $N$ is plotted against $1-{\lambda }_{\mathrm{av}}$, where ${\lambda }_{\mathrm{av}}$ is the average quantum Chernoff bound. In (c) and (d), the photon flux $N_f$ multiplied by the total time $t$ is plotted against $1-{\lambda }_{\mathrm{av}}$. The individual figures are tomography for (a) a single atomic qubit, (b) two atomic qubits, (c) a single photonic qubit with a single-detector configuration, and (d) two photonic qubits using single-detector configurations.