Experimental adaptive Bayesian tomography

We report an experimental realization of an adaptive quantum state tomography protocol. Our method takes advantage of a Bayesian approach to statistical inference and is naturally tailored for adaptive strategies. For pure states, we observe close to $N^{-1}$ scaling of infidelity with overall number of registered events, while the best nonadaptive protocols allow for $N^{-1/2}$ scaling only. Experiments are performed for polarization qubits, but the approach is readily adapted to any dimension.

Figure 1

Simulated tomography using three measurement selection methods: randomly sampled (red continuous line), MUBs (blue $\times$), and fully adaptive Bayesian tomography (black $\circ$). For these methods, the true state is random and pure, and the results presented here are the average of 20 independent runs. Overlaid dashed lines indicate the power-law fit. The functions $1-F=N^{-1/2}$ (magenta, dash-dotted line) and $1-F=N^{-1}$ (cyan, dashed line) are shown for comparison. To account for the state dependence of MUB tomography, we also present its performance for the “worst” and “best” (see text) true states (dark green $+$ and light green $•$, respectively).

Figure 2

Experimental setup. An attenuated laser is used as a source. The polarization state is prepared by a custom wave plate and analyzed by a sequence of quarter- and half-wave plates, followed by a polarizing beam splitter and two single-photon counters. Wave plates are rotated by electronically controlled step-motor drivers to allow for adaptivity.

Figure 3

Experimental results: mean infidelity $1-\widehat{F}(\rho ,\widehat{\rho })$ with current Bayesian mean estimate $\widehat{\rho }$ for random measurements (red, upper line), adaptive measurements (black, lower line), and measurements in MUBs [blue (dark gray) points]. Data points are averages over 10 experimental runs; shaded areas and error bars show the standard deviation of the mean. Dashed straight lines are power-law fits.

Figure 4

Experimental results: mean infidelity $1-\widehat{F}(\rho ,\overline{\rho })$ with “true” state $\overline{\rho }$ for random measurements (red, middle line), adaptive measurements (black, lower line), and measurements in MUBs [blue, upper line]. Data points are averages over 10 experimental runs; shaded areas show the standard deviation of the mean. Dashed straight lines indicate the power-law fits.