Realization of Single-Qubit Positive-Operator-Valued Measurement via a One-Dimensional Photonic Quantum Walk

We perform generalized measurements of a qubit by realizing the qubit as a coin in a photonic quantum walk and subjecting the walker to projective measurements. Our experimental technique can be used to realize, photonically, any rank-1 single-qubit positive-operator-valued measure via constructing an appropriate interferometric quantum-walk network and then projectively measuring the walker’s position at the final step.

Figure 1

Experimental schematic. (a) Detailed sketch of the setup for realization of unambiguous state discrimination of two equally probable single-qubit states via a three-step QW. Single photons are created via SPDC in a BBO crystal. One photon in the pair is detected to herald the other photon, which is injected into the optical network. (b) Setup for realization of a qubit SIC POVM via a five-step QW. The initial coin states for realization of SIC POVM $|{\psi }^{1,2}⟩$ are prepared by a HWP, whereas $|{\psi }^{3,4}⟩$ are prepared by a QWP with certain setting angles. Site-dependent coin flipping is realized by HWPs (and QWPs) with different setting angles placed in different optical modes. (c) Detailed interferometric setup formed by the third and fourth BDs, which are used in the experimental realization of a qubit SIC POVM. Unmounted WPs are inserted into specific optical paths such that only certain beams transit. WP angles are fixed (not rotatable) and calibrated via tomography before insertion.

Figure 2

Experimental data for unambiguous state discrimination via a photonic QW. Measured position distributions for the three-step QW with site-dependent coin and (a) initial coin state $|{\varphi }^{+}⟩$ and (b) $|{\varphi }^{-}⟩$; various coefficients $\varphi$ of $|{\varphi }^{\pm }⟩$ for unambiguous state discrimination. (c) Measured probability ${\eta }_{\mathrm{err}}$ for inconclusive results vs parameters $\varphi$, which are related to the state to be discriminated; compared to theoretical predictions. Error bars are smaller than portrayed by the symbols. (d) Position distribution for the three-step QW with initial coin state $|H⟩$, which is an equally weighted superposition of $|{\varphi }^{\pm }⟩$ with $\varphi =45°$. The blue and red bars show the experimental data and theoretical predictions, respectively. Error bars indicate the statistical uncertainty.

Figure 3

Experimental data of a qubit SIC POVM via a photonic QW. Measured probability distributions of the five-step QW with the site-dependent coin rotations and four different initial coin states $|{\psi }^i⟩$ with $i=1,2,3,4$ in (a)–(d), respectively.