Properties of dimension witnesses and their semidefinite programming relaxations

In this paper we develop a method for investigating semi-device-independent randomness expansion protocols that was introduced in Li et al. [H.-W. Li, P. Mironowicz, M. Pawłowski, Z.-Q. Yin, Y.-C. Wu, S. Wang, W. Chen, H.-G. Hu, G.-C. Guo, and Z.-F. Han, Phys. Rev. A 87, 020302(R) (2013)]. This method allows us to lower bound, with semi-definite programming, the randomness obtained from random number generators based on dimension witnesses. We also investigate the robustness of some randomness expanders using this method. We show the role of an assumption about the trace of the measurement operators and a way to avoid it. The method is also generalized to systems of arbitrary dimension and for a more general form of dimension witnesses than in our previous paper. Finally, we introduce a procedure of dimension witness reduction, which can be used to obtain from an existing witness a new one with a higher amount of certifiable randomness. The presented methods find an application for experiments [J. Ahrens, P. Badziag, M. Pawlowski, M. Zukowski, and M. Bourennane, Phys. Rev. Lett. 112, 140401 (2014)].

Figure 1

Lower bounds via SDP on the certified randomness for a dimension witness obtained from the CGLMP inequality [see Eq. (15) in Sec. 4] using the methods from Sec. 3a for different values of the dimension $d$.

Figure 2

Lower bounds via SDP on min-entropy certified by different Bell inequalities for different levels of noise. Several situations are considered. This illustrates the relations summarized in lemma 3b. Reduced Bell inequalities refer to lower bounds on reduced symmetric dimension witnesses (see Sec. 3c) for strategy P (see Sec. 3b). We observe that using reduced dimension witnesses provides an advantage in terms of certifiable randomness. Recall that by theorem 3a and a remark below it, the lower bound for the randomness of a dimension witness is given by the randomness of the Bell inequality minus ${log}_{2}d$. This plot refers to the case with $d=2$, and thus these methods that give values below ${log}_{2}2=1$ are not feasible for the given value of $p$. The reduced Bell operators are given by formulas (19), (22), (24), and (26), while Bell operators that may be used for lower bounding the randomness full symmetric dimension witnesses are given by formulas (18), (23), (25), and (29). These refer to $T2$ (2a), $T3$ (2b), $BC3$ (2c), and $\text{modCHSH}$ (2d) cases, respectively.

Figure 3

Numerical lower bounds via SDP and upper bounds on the randomness certified for strategy P (see Sec. 3b), for both reduced (see Sec. 3c) and full certificates and different Bell operators and dimension witnesses. Here we observe even more advantage for the reduced versions. The plots refer to the cases of $T2$ (3a), $T3$ (3b), $BC3$ (3c), and $\text{modCHSH}$ (3d), respectively.

Figure 4

Lower bounds via SDP on the certified randomness in the semi-device-independent scenario for the reduced dimension witnesses (see Sec. 3c) when the untrusted vendor uses the mixed strategy with different values of the parameter $\delta$ (see Sec. 3b). If a certain value of a dimension witness is impossible to achieve with a given $\delta$, then, since the eavesdropper cannot mislead us this way, we use the value 1. The cases of $T2$ (4a), $T3$ (4b), $BC3$ (4c), and $\text{modCHSH}$ (4d) are shown on these plots.

Figure 5

Lower bounds via SDP on the certified randomness for both reduced and symmetric dimension witnesses (see Sec. 3c) when the untrusted vendor uses the mixed strategy with the worst case value of the parameter $\delta$ (see Sec. 3b), which is 1 in all cases. The plots shows the cases referring to $T2$ (5a), $T3$ (5b), $BC3$ (5c), and $\text{modCHSH}$ (5d).