Robust amplification of Santha-Vazirani sources with three devices

We demonstrate that amplification of arbitrarily weak randomness is possible using quantum resources. We present a randomness amplification protocol that involves Bell experiments. We find a Bell inequality that can amplify arbitrarily weak randomness and give a detailed analysis of the protocol involving it. Our analysis includes finding a sufficient violation of the Bell inequality as a function of the initial quality of randomness. It has a very important property that for any quality the required violation is strictly lower than possible to obtain using quantum resources. Among other things, this means that the protocol takes a finite amount of time to amplify arbitrarily weak randomness.

Figure 1

Maximal bias of the measurement outcome as a function of the weakness of randomness and success probability of winning in a nonlocal game based on the tripartite Mermin inequality. This plot can be also understood as a critical value of the success probability required to take inputs from $\epsilon$-free source and get outcomes with bias less than ${\epsilon }^{\prime }$.

Figure 2

Sufficient winning probability for randomness amplification in a nonlocal game based on the tripartite Mermin inequality as a function of the sources' freedom. If $P_s$ is above the plot it means that the bias of the final bit $y$ is lower than that of the Santha-Vazirani (SV) source. Note that higher values of $P_s$ are required for extremal initial $\epsilon$. This is because if $\epsilon$ is large, the initial quantity of randomness is bad, which makes the amplification difficult. On the other hand, if $\epsilon$ is already low, then to amplify it we need to obtain an even more random bit, which is again difficult.

Figure 3

Schematic diagram of the randomness amplification protocol.