Experimental adaptive quantum tomography of two-qubit states

We report an experimental realization of adaptive Bayesian quantum state tomography for two-qubit states. Our implementation is based on the adaptive experimental design strategy proposed in the work by Huszár and Houlsby [F. Huszár and N. M. T. Houlsby, Phys. Rev. A 85, 052120 (2012).] and provides an optimal measurement approach in terms of the information gain. We address the practical questions which one faces in any experimental application: the influence of technical noise and the behavior of the tomographic algorithm for an easy-to-implement class of factorized measurements. In an experiment with polarization states of entangled photon pairs, we observe a lower instrumental noise floor and superior reconstruction accuracy for nearly pure states of the adaptive protocol compared to a nonadaptive protocol. At the same time, we show that for the mixed states, the restriction to factorized measurements results in no advantage for adaptive measurements, so general measurements have to be used.

Figure 1

Experimental setup. The entangled photon pairs are generated in two BBO crystals with perpendicular axes, pumped by linearly polarized light from a 408 nm diode laser. Projective measurements are performed separately for each photon of the pair by means of half- and quarter-wave plates followed by Wollaston prisms.

Figure 2

Experimental results. The experimentally obtained dependencies of the posterior distribution size ${\overline{d}}_B^2$ on the overall number of detected pairs for the cases of a (a) separable and (b) maximally entangled polarization state. The dot-dashed lines are power-law fits to the data, and the dashed gray lines are the dependencies $\propto N^{-1/2}$ and $\propto N^{-1}$ shown here and on consecutive plots for comparison. Legend: FR stands for factorized random measurements and FA stands for factorized adaptive measurements.

Figure 3

Experimental tomography in the presence of instrumental errors. The dependence of the spread of experimental results (see text for definition) on the overall number of registered events. Dashed lines correspond to random protocol, while solid ones correspond to adaptive tomography. Color encodes different amounts of additional instrumental noise in the experimental run: no additional noise (blue), $3^{\circ }$ (dark yellow), and $5^{\circ }$ (green) of additional wave-plate uncertainty. The shaded areas correspond to one standard deviation for the ensemble of 10 different tomography runs.

Figure 4

The results of numerical simulations for pure states. (a) The dependence of distance to the true state averaged over the posterior distribution on the total number of registered events $N$ and (b) histograms for the values of $N$ required to reach a fixed accuracy $d_B^2={10}^{-3}$ are shown. Legend: F stands for factorized, G stands for general, R stands for random, and A stands for adaptive measurements.

Figure 5

The results of numerical simulations for mixed states picked at random from the uniform distribution with respect to the Bures-induced measure. (a) The dependence of a distance to the true state averaged over the posterior distribution on the total number of registered events $N$ and (b) histograms for the values of $N$ required to reach a fixed accuracy $d_B^2={10}^{-3}$ are shown.

Figure 6

Tomography performance for states of different purity. The value of $N$ required to reach the specified precision, measured by the distance to the true state of ${\overline{d}}_B^2\left(N\right)=d_{\mathrm{err}}^2=5\times {10}^{-3}$ averaged within the given purity layer, is shown. The stars correspond to experimental points, while dots are the results of numerical simulation. Error bars show a standard deviation for averaged values of $N$. The solid lines are guides to the eye.

Figure 7

Influence of the prior distribution. The dependence of the average distance to the true state $d_B^2(\widehat{\rho }\left(N\right),{\rho }_0)$ on the number of registered events $N$ for the FA protocol with Bures (blue line) and uniform in simplex (yellow line) prior. For the GA protocol, the problem with convergence is less visible (red line). The distance of the maximum-likelihood estimate to the true state (green line) is shown for comparison. The shaded areas correspond to one standard deviation for the ensemble of 10 different tomography runs.

Figure 8

The average ratio $R_{dd}$ of the Bures distance to the true state $d_B^2(\widehat{\rho }\left(N\right),{\rho }_0)$ and the posterior size ${\overline{d}}_B^2\left(N\right)$ in Bures metric. Simulated results for GA and FA protocols and different number of samples used in the sequential Monte Carlo calculation are shown. It is clear that for a sufficient number of samples (which is lower for the GA protocol), the posterior distribution size is a reasonable estimate of the tomographic accuracy.