Generalized Quantum Measurements on a Higher-Dimensional System via Quantum Walks

Quantum measurements play a fundamental role in quantum mechanics. Especially, generalized quantum measurements provide a powerful and versatile tool to extract information from quantum systems. However, how to realize them on an arbitrary higher-dimensional quantum system remains a challenging task. Here we propose a simple recipe for the implementation of a general positive-operator valued measurement (POVM) on a higher-dimensional quantum system via a one-dimensional discrete-time quantum walk with a two-dimensional coin. Furthermore, using single photons and linear optics, we realize experimentally a symmetric, informationally complete (SIC) POVM on a three-dimensional system with high fidelity. As an application, we realize a qutrit state tomography with SIC-POVM and confirm that the infidelity scaling achieved by the qutrit SIC-POVM is as good as that based on mutually unbiased bases. Our study paves the way to explore physics and information in higher-dimensional quantum systems and finds applications in various quantum information processing tasks that rely on generalized quantum measurements.

Figure 1

(a) Illustration of the algorithm for the generation of an arbitrary rank-1 POVM on a qudit. Coin operations $R_{j,k}$ for $k=1,\dots ,r_j$. (b) Realization of a qutrit SIC-POVM via a 15-step QW. The coin qubit and the walker in positions are taken as a qutrit of interest, whereas the other positions of the walker act as ancillae. Position-dependent coin operations are specified based on the algorithm. Some coins at certain positions are not shown as those positions are not reachable by a QW from the used initial state. The 9 detectors $E_i$ correspond to the 9 outcomes of the qutrit SIC-POVM.

Figure 2

Schematic of the experimental setup. Heralded single photons are generated via type-I spontaneous parametric down-conversion and prepared in 9 different qutrit states vis a beam displacer (BD) and wave plates (${\mathrm{H}}_1$, ${\mathrm{Q}}_1$, ${\mathrm{H}}_2$, and ${\mathrm{Q}}_2$). Combined with the first coin operation, the basis states are encoded by the coin and the walker in positions 0 and 2. Then the single photons undergo a POVM device based on a QW with a two-dimensional coin, which includes coin operations realized by wave plates and conditional shift operations realized by BDs. Finally, photons are detected by avalanche photodiodes (APDs) via a coincidence with the trigger photons. Each of the 9 APDs corresponds to an outcome of a POVM element.

Figure 3

Probabilities of the photons being detected at each APD after the POVM performed on the basis state $|{\phi }_i⟩$ with $i=1,\dots ,9$. Solid and hollow bars represent the experimental results and their theoretical predictions, respectively. Error bars indicate the statistical uncertainty, obtained via error propagation assuming Poissonian statistics.

Figure 4

Matrix forms of the elements $E_j$ of a qutrit SIC-POVM. Real and imaginary parts of the matrix forms of the 9 reconstructed POVM elements are plotted by solid bars, while hollow bars represent their theoretical predictions.

Figure 5

Infidelity versus $N$ for two different measure methods: a qutrit SIC-POVM and MUBs. We choose three different states: $|{\phi }_1⟩$, $|{\phi }_4⟩$, and $|{\phi }_8⟩$, for examples. Both experimental and numerical results for SIC-POVM and numerical results for MUBs are shown for comparison. Experimental and numerical results are both shown. Numerical results are obtained by Monte Carlo simulations. Each data point is the average of 1000 repetitions.