Reduced gap between observed and certified randomness for semi-device-independent protocols

Quantum systems can be used to certify the existence of true randomness in nature, and semi-device-independent (SDI) randomness expansion has been used to achieve this goal under the prepare-measure configuration. Usually, randomness is certified with the degree of violation of some inequalities. However, the observed statistics characterizing SDI protocols contain much richer information than just a single value of the inequality. Thus, there might be a gap between the randomness in observed statistics and the randomness we certify. The purpose of this paper is to optimize the use of the full observed statistics to reduce that gap. It shows that certifying randomness directly from the full observed statistics but not some inequality is available in the SDI protocols. Besides, average of guessing probabilities should be considered as randomness quantitative standards, since it certifies more randomness than the previous maximal guessing probability method. In the analysis, we take into account the assumption that the devices and the potential adversary are classically correlated.

Figure 1

The SDI-protocol based on $2\to 1$ QRAC test, which concerns state preparation-measurement black boxes. Both of the boxes are in the same safe laboratory.

Figure 2

The relationship between the violation value $Q$ and maximal guessing probability $G_{\max }$, where $G_p$ is the observed maximal guessing probability in $P,G_{xy} \left(G_{xy,Q}\right)$ is the certified maximal guessing probability in $P$ using the $W$ inequality, and $G_{xy,S}$ is the certified maximal guessing probability in $P$ directly from the full observed statistics. The curves of $G_{xy,Q}$ and $G_{xy,S}$ have taken into account the classical side information.

Figure 3

The relationship between the violation value $Q$ of $W$ and maximal average guessing probability ${\tilde{G}}_i$ without classical side information, where the curves of ${\tilde{G}}_3$ and ${\tilde{G}}_4$ are coincident.

Figure 4

The difference between ${\tilde{G}}_{4,S}^{\prime }$ and ${\tilde{G}}_{4,Q}^{\prime }$ which get from the full observed statistics directly and the $W$ inequality, respectively. The results have taken into account the classical side information. The curves of ${\tilde{G}}_{4,Q}^{\prime }$ and ${\tilde{G}}_4$ shown in Fig. 3 are coincident.