Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion

We consider 1-qubit mixed quantum state estimation by adaptively updating measurements according to previously obtained outcomes and measurement settings. Updates are determined by the average-variance–optimality (A-optimality) criterion, known in the classical theory of experimental design and applied here to quantum state estimation. In general, A optimization is a nonlinear minimization problem; however, we find an analytic solution for 1-qubit state estimation using projective measurements, reducing computational effort. We compare numerically the performances of two adaptive and two nonadaptive schemes for finite data sets and show that the A-optimality criterion gives more precise estimates than standard quantum tomography.

Figure 1

Sketch of a generic procedure for an adaptive quantum estimation scheme.

Figure 2

Average expected loss ${\overline{\Delta }}_N^{\mathrm{ave}}(u^N,{\mathit{s}}^{\mathrm{est}})$ against the number of measurement trials $N$: ${\overline{\Delta }}_N^{\mathit{HS}\mathrm{ave}}$ integrated via the Bures distribution ${\mu }_{\mathrm{Bures}}$ (HS-Bures), ${\overline{\Delta }}_N^{\mathit{HS}\mathrm{ave}}$ integrated via the Euclidean distribution ${\mu }_{\mathrm{Euclid}}\left(\mathit{s}\right)=3/4\pi$ (HS-Euclid), ${\overline{\Delta }}_N^{\mathit{IF}\mathrm{ave}}$ integrated via ${\mu }_{\mathrm{Bures}}$ (IF-Bures), and ${\overline{\Delta }}_N^{\mathit{IF}\mathrm{ave}}$ integrated via ${\mu }_{\mathrm{Euclid}}$ (IF-Euclid). The dashed spaced (orange) line in (IF-Bures) is the bound of separable (including adaptive) schemes derived in Ref. [14]. The number of measurement trials $N_{\max }$ is 1000, the number of sequences used for the calculation of the statistical expectation values $N_{\mathrm{mean}}$ is 1000, and the number of sample points used for the Monte Carlo integration $N_{\mathit{MC}}$ is 3200.

Figure 3

Pointwise expected loss ${\overline{\Delta }}_N(u^N,{\mathit{s}}^{\mathrm{est}}|\mathit{s})$ against the number of trials $N$ (the horizontal and vertical axes are both logarithmic scale): (HS-P1), (HS-P2), and (HS-P3) are the expected squared Hilbert-Schmidt distances for $\mathit{s}$ given by $(r,\theta ,\phi )=(0,0,0),(0.99,0,0),(0.99,\pi /4,\pi /4)$, and (IF-P1), (IF-P2), and (IF-P3) are the expected infidelities for the same three true states, respectively. The number of measurement trials $N_{\max }$ is $10000$, and the number of sequences used for the calculation of statistical expectation values $N_{\mathrm{mean}}$ is 1000.

Figure 4

Purity dependence of average expected infidelity at $N=1000$. Cross (black) for AHS, saltire (red) for AIF, asterisk (blue) for $XYZ$, and square (green) for URS. The number of sequences used for the calculation of the statistical expectation values $N_{\mathrm{mean}}$ is 1000, and the number of sample points used for the Monte Carlo integration $N_{\mathit{MC}}$ is 500 for each Bloch radius $r$.

Figure 5

Distribution of measurement Bloch vectors at $N=100$ (left column), 1000 (middle column), and $10000$ (right column) for 900 runs; The true state is $(r,\theta ,\phi )=(0.99,\pi /4,\pi /4)$. The upper three plots (AHS-100), (AHS-1000), and (AHS-10000) are AHS while the lower three (AIF-100), (AIF-1000), and (AIF-10000) are AIF.

Figure 6

An implementation of adaptive projective measurements for single-photon polarization qubits in quantum optics. HWP and QWP are half- and quarter-wave plates, PBS is a polarization beam splitter, PD are photodetectors, and CC denotes classical computation. The direction of the projective measurement are adapted by changing the waveplate angles.

Figure 7

Purity dependence of average expected infidelity of $XYZ$ and URS schemes: The average is taken over all directions $\theta$ and $\phi$ for each Bloch radius $r$. Average expected infidelity of $XYZ$ repetition (left) and URS (right) for different Bloch radii. Solid line (black) is for $r=0$, dotted line (green) is for $r=0.7$, dotted spaced line (blue) is for $r=0.9$, dashed line (light blue) is for $r=0.93$, dashed spaced line (purple) is for $r=0.97$, and dotted dashed line (red) is for $r=0.99$. The number of sequences used for the calculation of the statistical expectation values $N_{\mathrm{mean}}$ is 1000, and the number of sample points used for the Monte Carlo integration $N_{\mathit{MC}}$ is 500 for each Bloch radius $r$.