Experimental investigation of partially entangled states for device-independent randomness generation and self-testing protocols

Previous theoretical works showed that all pure two-qubit entangled states can generate one bit of local randomness and can be self-tested through the violation of proper Bell inequalities. We report an experiment in which nearly pure partially entangled states of photonic qubits are produced to investigate these tasks in a practical scenario. We show that small deviations from the ideal situation make low entangled states impractical to self-testing and randomness generation using the available techniques. Our results show that in practice lower entanglement implies lower randomness generation, recovering the intuition that maximally entangled states are better candidates for device-independent quantum information processing.

Figure 1

Experimental setup used for randomness certification and self-testing. See more details in the main text.

Figure 2

Purity $\mathrm{Tr}\left({\rho }_t^2\right)$ (blue points) and the fidelity (orange points) with respect to the closest PES obtained from quantum state tomography. In the horizontal axis the angle $\theta$ characterizing the closest PES. The error bars are obtained with Gaussian error propagation and considering the Poisson statistics of the recorded coincidence counts.

Figure 3

Observed violation of the tilted Bell inequality. The orange lower line corresponds to the local bound, above which nonlocality can be proven. The green upper line is the maximal violation that can be achieved, while the blue dots are the experimental values obtained. The error bars lie within the experimental dots and are obtained with Gaussian error propagation and considering the Poisson statistics of the recorded coincidence counts.

Figure 4

Experimental randomness certification as a function of entanglement. Almost all the cases can certify randomness. However, they are far from, in principle, the possible value of 1 bit of randomness. The solid line is given as a reference, and considers PES with 0.5% of white noise. The error bars are obtained with Gaussian error propagation and considering the Poisson statistics of the recorded coincidence counts.

Figure 5

Fidelity bounds with respect to target PES. The solid orange line, added as a reference, shows the fidelity of a partially entangled state and a fully separable ones, i.e., $F(|00\rangle \langle 00|,|\psi \left(\theta \right)\rangle \langle \psi \left(\theta \right)|)={cos}^2\left(\theta \right)$. In the horizontal axis the angle $\theta$ that is self-tested by the fidelity bound. Notice that for the least entangled state the method self-tests a state which is below the separable bound and renders an angle which is very far from what is obtained through tomography (see Table 1). The error bars are obtained with Gaussian error propagation and considering the Poisson statistics of the recorded coincidence counts.