Experimental quantum tomography assisted by multiply symmetric states in higher dimensions

High-dimensional quantum information processing has become a mature field of research with several different approaches being adopted for the encoding of $D$-dimensional quantum systems. Such progress has fueled the search of reliable quantum tomographic methods aiming for the characterization of these systems, most of these methods being specifically designed for a given scenario. Here, we report on a tomographic method based on multiply symmetric states and on experimental investigations to study its performance in higher dimensions. Unlike other methods, it is guaranteed to exist in any dimension and provides a significant reduction in the number of measurement outcomes when compared to standard quantum tomography. Furthermore, in the case of odd dimensions, the method requires the least possible number of measurement outcomes. In our experiment we adopt the technique where high-dimensional quantum states are encoded using the linear transverse momentum of single photons and are controlled by spatial light modulators. Our results show that fidelities of $0.984\pm 0.009$ with ensemble sizes of only $1.5\times {10}^5$ photons in dimension $D=15$ can be obtained in typical laboratory conditions, thus showing its practicability in higher dimensions.

Figure 1

Left panels: Condition number $\mathcal{C}(\mathcal{G}\left(\alpha \right))$ as function of $\alpha$—given that fiducial state $|{\alpha }_0\rangle$ is given by Eqs. (24) and (25)—for dimensions 6 and 15. Red squares indicate the values of $\alpha$ that were used in the experiment reported in this article. Since $\mathcal{C}(\mathcal{G}\left(\alpha \right))$ might adopt very different values, these graphs were presented in logarithmic scale. Right panels: Fidelity of reconstructed states simulated by Monte Carlo method.

Figure 2

Experimental setup. At the preparation state we generate an arbitrary state of a single qudit that is encoded in a single photon. A 690-nm continuous-wave laser, an AOM, and calibrated attenuators (not shown in figure for sake of clarity), are employed to generate weak coherent pulses. These illuminate spatial light modulators SLM1 and SLM2, which with the help of polarizers (P) and quarter-wave plates (QWP) are employed to modulate the incoming light in amplitude and phase, correspondingly. Electronically addressable slit patterns at SLM1 and SLM2 control the state $|{\mathrm{\Psi }}^d\rangle$ of the photonic qudit. The measurement stage projects state $|{\mathrm{\Psi }}^d\rangle$ onto a predefined, arbitrary single-qudit state $|{\alpha }_{sj}\rangle$. Spatial light modulators SLM3 and SLM4, combined with a pointlike avalanche photodiode (APD), implement the projection. In this way we are able to estimate the probability $|\langle {\alpha }_{sj}|{\mathrm{\Psi }}^d\rangle {|}^2$. A set of lenses is employed to transport the images generated by the SLMs along the setup. Focal lengths are as follows: L1 = 25 mm, L2 = 200 mm, L3 = L4 = L5 = L6 = L7 = L8 = 125 mm, and L9 = 100 mm.

Figure 3

Examples of the Monte Carlo simulations performed (upper panels) and their respective histograms (lower panels). Three horizontal dot-dashed lines in the upper graphs represent the mean value ($\langle F\rangle$) of the simulations and the $\langle F\rangle \pm 5\sigma$ interval. The continuous line in each histogram represents a fitted beta distribution. For each fitted function, the probability of having a value outside the $\pm 5\sigma$ interval is $\sim {10}^{-6}$.

Figure 4

Reconstructed quantum states for $D=6$. The insets show the theoretically expected results.

Figure 5

Reconstructed quantum states for $D=15$. The insets show the theoretically expected results.