Experimental demonstration of optimal measurement for unambiguously discriminating asymmetric qudit states

Abstract

Identification of nonorthogonal quantum states without error is crucial for various applications in quantum information technology, as well as the foundations of quantum physics. Theoretical studies have proposed measurements that maximize the success probability of unambiguously discriminating quantum states. However, these methods are not always experimentally feasible, which has led most demonstrations to focus on equiprobable symmetric states. Here, we establish a projective measurement scheme that optimally discriminates multiple asymmetric qudit states. We experimentally demonstrate this optimal projective measurement using a photonic orbital angular momentum state, where asymmetric qudit states are encoded in the Laguerre-Gaussian modes of a heralded single-photon state. Our results have broad applications in high-dimensional quantum state-based quantum information processing, including quantum key distribution and quantum sensing.

Figures (9)

• Figure 1: Concept of USD, where Alice prepares one of three qutrit states. The two angles $\varphi$ and $\theta$ in Alice's quantum states are distinct, indicating asymmetry. Alice prepares a quantum state $|\psi_x\rangle$, and Bob performs a measurement $\mathcal{M}$ to identify the prepared state. There are only two possible outcomes in USD. If the measurement outcome is conclusive ($y\not=?$) with a probability $p_{\rm succ}$, Bob can confidently determine that Alice prepared $|\psi_x\rangle$. Otherwise, with probability $p_{\rm fail}=1-p_{\rm succ}$, the measurement outcome provides no useful information.
• Figure 2: Required dimension of a projective measurement for establishing a generalized measurement that optimally discriminates $d$ high-dimensional qudit states. Here, both red and blue bars show the dimension of a state space and that of the projective measurement, respectively.
• Figure 3: Experimental setup for unambiguously discriminating three asymmetric qutrit states. A PPKTP crystal is pumped by a 405 nm pump laser, generating a pair of single photons via the via the type-II spontaneous parametric down-conversion process (SPDC). The idler photon is sent to APD1 for heralding, while the signal photon is directed to APD2 through an single-mode optical fiber. SLM1 is used to generate an asymmetric photon state $|\psi_j\rangle$, and SLM2 is used to discriminate the prepared states (H, Half-wave plate; Q, Quarter-wave plate; P, Polarizing beam-splitter; L, Lens; LPF, Long pass filter; SLM, Spatial light modulator; APD, Avalanche photodiode; TCSPC, Time-correlated single-photon counter).
• Figure 4: Experimentally obtained probability distribution for $\theta = \frac{2\pi}{3}, \xi = \frac{\pi}{3}$ and each $\varphi$. (a) $\varphi=0.53\pi$, (b) $\varphi=0.6\pi$, (c) $\varphi=0.66\pi$ and (d) $\varphi=0.76\pi$. Each measurement probability $|\langle D_k|\psi_j\rangle|^2$ is colored with blue, red, and green bars corresponding to success, error, and failure cases, respectively.
• Figure 5: Experimental results on success probability considering different prior probabilities for each $\varphi$. Note that the prepared states are symmetric only when $\varphi = \frac{2}{3}\pi$. Red, blue, and green dots (lines) indicate experimental (theoretical) values for the cases of $\{q_1,q_2,q_3\}=\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\}$, $\{\frac{5}{12},\frac{7}{24},\frac{7}{24}\}$, and $\{\frac{1}{2}, \frac{1}{4}, \frac{1}{4}\}$, respectively. (a) The success probability calculated from the experimental data. (b) The error probability from the experimental data, compared with the MESD bound. The error bars in (a) and (b) represent the standard deviation $\sigma=1$, obtained from 1000 repetitions of the Monte Carlo simulation with Poissonian errors.