Robustness of quantum-randomness expansion protocols in the presence of noise

In this paper we investigate properties of several randomness generation protocols in the device-independent framework. Using Bell-type inequalities it is possible to certify that the numbers generated by an untrusted device are indeed random. We present a selection of certificates which guarantee two bits of randomness for each run of the experiment in the noiseless case and require the parties to share a maximally entangled state. To compare them we study their efficiency in the presence of white noise. We find that for different amounts of noise different operators are optimal for certifying most randomness. Therefore, the vendor of the device should use different protocols depending on the amount of noise expected to occur. Another of our results that we find particularly interesting is that using a single Bell operator as a figure of merit is rarely optimal.

Figure 1

Lower bounds on min-entropies for protocols described in Sec. 2a (based on Braunstein-Caves inequalities) as a function of noise.

Figure 2

Lower bound on min-entropy for protocol described in Sec. 2b ($E_0$ with $E_1$) as a function of $\varphi$ [see Eq. (2.4)] for different noises.

Figure 3

Lower bound on min-entropy for protocol described in Sec. 2c ($T3C$) as a function of $C$ (see text) for different noises.

Figure 4

Lower bounds on min-entropies for protocols described in Secs. 2d1 (modified CHSH) and 2d2 (other randomly generated protocols) as a function of noise.

Figure 5

Comparison of the three most efficient of investigated protocols for $0.84\le p\le 0.94$.