Experimental Demonstration of Adaptive Quantum State Estimation

The first experimental demonstration of an adaptive quantum state estimation (AQSE) is reported. The strong consistency and asymptotic efficiency of AQSE have been mathematically proven [A. Fujiwara, J. Phys. A 39, 12489 (2006)]. In this Letter, the angle of linear polarization of single photons, the phase parameter between the right and the left circularly polarization, is estimated using AQSE, and the strong consistency and asymptotic efficiency are experimentally verified. AQSE will provide a general useful method in both quantum information processing and metrology.

Figure 1

Schematic of adaptive quantum state estimation. Photons are linearly polarized with a polarization direction determined by HWP0. The polarization is analyzed by HWP1 and the polarizing beam splitter (PBS). The controller sets HWP1 to an angle calculated on the basis of the photon measurement results.

Figure 2

(a) Schematic of the experimental setup. (b)(c) An example showing the update of a log-likelihood function. The second term $\log ⟨\psi (\theta )|M(x_n;{\widehat{\theta }}_{n-1})|\psi (\theta )⟩$ in Eq. (6) is shown in panel (b), and the updated $l_n(\theta )$ is shown in panel (c). The (blue) arrows in (b) and (c) indicate the true value ${\theta }^t$.

Figure 3

(a) Trajectories of estimated HWP0 angles against the number $n$ of photons for $r=500$ repetitions is shown in a three-dimensional plot. (b) The first 10 curves are superposed in a two-dimensional graph.

Figure 4

Histogram of the observed data obtained by $r=500$ independent experiments of adaptive estimation scheme, each using $n=300$ photons. These histograms were taken for four different true values of (a) 0 deg, (b) 30 deg, (c) 60 deg, and (d) 78.3 deg.