Fig. 1. Standard randomness certification scenario in device-independent ways. An entangled source, two measurement stations, Alice and Bob, and an additional observer, Eve. The source simultaneously emits particles to two measurement stations, Alice and Bob. Each of them randomly performs the local measurement setting $x$ or $y$ and obtains outcome $a$ or $b$, respectively. The observed correlation is represented by the conditional probability $P(a,b|x,y)$. From the perspective of security, we will assume that Eve might be able to guess the outcomes of Alice’s/Bob’s measurement.

Fig. 2. Schematic of our experimental setup for randomness certification based on SIC-POVM. (a) A maximally entangled state $|{\mathrm{\Psi }}_{\mathrm{AB}}\rangle =(|\mathrm{HH}\rangle -|\mathrm{VV}\rangle )/\sqrt{2}$ is generated with type-II SPDC sources pumped by pulsed lasers. (b) A four-outcome POVM is implemented by employing five-step quantum walks. (c) Projective measurement is implemented with a QWP, an HWP, and a PBS. BBO, $\beta$-barium borate crystal; BPF, band pass filter; C-BBO, sandwich-type $\mathrm{BBO}+\mathrm{HWP}+\mathrm{BBO}$ combination; QWP, quarter-wave plate; HWP, half-wave plate; PBS, polarizing beam-splitter; ${\mathrm{LiNbO}}_3$, lithium niobate crystal, which is used for spatial compensation; ${\mathrm{YVO}}_4$, yttrium orthovanadate crystal, which is used for temporal compensation; BD, beam displayer; CL, collimation lens.

Fig. 3. Bloch vector of SIC-POVM. The tetrahedron formed by the dotted black line represents the initial SIC-POVM, and the tetrahedron formed by the solid red line represents the target SIC-POVM.

Fig. 4. Tomography of the prepared maximally entangled state. The real and imaginary parts are shown in the left and right panels, respectively.

Table 1. Theoretical and Experimental Results of the Elegant Bell Inequality

Table 2. Theoretical and Experimental Values for the Probabilities of the Four Outcomes of SIC-POVM