Device- and semi–device-independent random numbers based on noninequality paradox

In this work, we propose device-independent true random number expansion protocols based on noninequality paradoxes such as Hardy's and Cabello's nonlocality arguments, thus highlighting the noninequality paradox as an important resource for device-independent quantum-information processing, in particular for generating true randomness. As a byproduct, we find that the nonlocal bound of the Cabello argument with arbitrary dimension is the same as the one achieved in the qubits system. More interestingly, we propose a dimension witness paradox based on the Cabello argument which can be used for constructing a semi–device-independent true random number expansion protocol.

Figure 1

A lower bound on min-entropy $H_{\infty }(a,b|A_0,B_0)$ as a function of Hardy's parameter $p_{\mathrm{Hardy}}$. Lower bound means we apply the global optimization calculation method, which reach the minimal value of $H_{\infty }(a,b|A_0,B_0)$.

Figure 2

A lower bound on min-entropy $H_{\infty }(a,b|A_0,B_0)$ as a function of $\epsilon$.

Figure 3

A lower bound on min-entropy $H_{\infty }(a,b|A_0,B_0)$ as a function of Cabello's parameter $p_{\mathrm{Cabello}}$.

Figure 4

A lower bound on min-entropy $H_{\infty }(a,b|A_0,B_0)$ for $K=2$ as a function of parameter $p_{\mathrm{Hardy}}$.

Figure 5

A lower bound on min-entropy $H_{\infty }(a,b|A_0,B_0)$ for $K=3$ as a function of parameter $p_{\mathrm{Hardy}}$.

Figure 6

A lower bound on min-entropy $H_{\infty }\left(b\right|A_0,a,B_0)$ as a function of parameter $p_{\mathrm{Cabello}}$.