Implementation of generalized measurements on a qudit via quantum walks

Quantum measurements play a fundamental role in quantum mechanics and quantum information processing, but it is not easy to implement generalized measurements, the most powerful measurements allowed by quantum mechanics. Here, we propose a simple recipe for implementing generalized measurements on a qudit via quantum walks. With this recipe, any discrete quantum measurement can be implemented via a one-dimensional discrete quantum walk; the number of steps is only two times the number of measurement outcomes. As an illustration, we present a unified solution for implementing arbitrary symmetric informationally complete measurements in dimension 3.

Figure 1

Schematic diagrams for implementing a rank-1 POVM on a qudit via a discrete quantum walk. Here, the target POVM $\{E_1,E_2,E_3,E_4\}$ has four elements and is realized using a three-iteration (six-step) quantum walk. The initial coin state at position $x=0$ is the qudit state of interest. The upper plot illustrates the algorithm presented in Sec. 4a, and the four detectors $E_1$–$E_4$ correspond to the four POVM elements, respectively. The detector $E_1$ ($E_2$) can also be put at position 2 after the first (second) iteration, as illustrated in the lower plot, because subsequently no coin operators are present along the corresponding paths.