Relationship between semi- and fully-device-independent protocols

We study the relation between semi- and fully-device-independent protocols. As a tool, we use the correspondence between Bell inequalities and dimension witnesses. We present a method for converting the former into the latter, and vice versa. This relation provides us with interesting results for both scenarios. First, we find random-number-generation protocols with higher bit rates for both the semi- and fully-device-independent cases. As a byproduct, we obtain classes of Bell inequalities and dimension witnesses. Then, we show how optimization methods used in studies on Bell inequalities can be adopted for dimension witnesses.

Figure 1

Schematic representation of (1) DI and (2) SDI protocols and of our method for finding the corresponding ones.

Figure 2

The lower bounds on $H_{\infty }(a,b|x,y)$ as functions of $P_n$.

Figure 3

Lower bound on the min entropy $H_{\infty }\left(b\right|a,x=1,y)$ as a function of coefficient $\alpha$ for the maximal quantum value of $W_{\alpha }$. In [16], a large amount of randomness is generated only for one setting of $x$. Here we observe the same result with high randomness for $x=1$ and low randomness for $x=0$.

Figure 4

Min entropy bounds for the SDI randomness-generation protocol based on the $2\to 1$ quantum random access code. The dots are obtained from the Levenberg-Marquardt algorithm used in [10], which is not guaranteed to find global minima, while the line depicts the SDP method described here. Note that state preparation in the SDI protocol assumes that $p\left(a\right|x,y)=\frac{1}{2}$.