Experimental realization and self-testing of semisymmetric informationally complete measurements via a one-dimensional photonic quantum walk

Abstract

Generalized quantum measurements play a crucial role in quantum mechanics, and symmetric informationally complete positive operator-valued measurements (SIC POVMs) provide a powerful and flexible framework for extracting information from quantum systems. However, the existence of SIC-POVMs in every finite dimension remains an open question, which has stimulated extensive research into alternative classes of POVMs. Recently, Geng $et$ $al$. [Phys. Rev. Lett. 126, 100401 (2021)] proposed a broader class of SIC POVM, called semisymmetric informationally complete POVM (semi-SIC POVM), which extends beyond SIC POVM. In this work, we focus on the four-outcome POVMs and experimentally realize the semi-SIC POVMs using a one-dimensional discrete-time quantum walk. Additionally, employing single photons and linear optics, we perform an experimental self-testing of semi-SIC POVMs in the semi-device-independent manner. Our results pave the way for exploring quantum certification with generalized quantum measurements.

Figures (4)

• Figure 1: Scenario considered for certifying the semi-SIC POVMs. (a) Alice prepares one of four different quantum states and sends to Bob, Bob performs either one of three different dichotomic measurements or a four-outcome povm measurement. (b) The Bloch vectors of semi-SIC POVMs for $B$ =1/12, 1/13, 1/14, and 1/15. When $B$ = 1/12, the semi-SIC POVM reduces to the SIC-POVM.
• Figure 2: Experimental setup. Semi-SIC POVMs are implemented via a five-step quantum walk. In a one-dimensional discrete-time quantum walk, the position of the walker and the coin state are encoded by the spatial modes and the polarization states of the single photons, respectively. The HWP1 and QWP1 are used to prepare the initial state. The POVM measurements consist of five beam displacers and ten wave plates. The flipping of the site-dependent coin is enabled with different HWPs and QWPs in the optical path. PBS, polarization beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; IF, interference filter; BD, beam displacer; DM, dichroic mirror.
• Figure 3: Experimental results of semi-SIC POVMs via a photonic quantum walk. The measured probability distributions for semi-SIC POVMs with $B = 1/12$, $1/13$, $1/14$, and $1/15$ show close agreement with theoretical predictions, with a maximum deviation of less than $0.005$.
• Figure 4: Experimental results for self-testing of semi-SIC POVMs. The witness values $W$ for $B$ = 1/13, 1/14, and 1/15 are $7.2908\pm0.0304$, $7.5615\pm0.0149$ and $7.9393\pm0.0434$, respectively. These experimental results are close to the theoretical predictions and the error bars indicate the statistical uncertainty.