Randomness requirement on the Clauser-Horne-Shimony-Holt Bell test in the multiple-run scenario

The Clauser-Horne-Shimony-Holt inequality test is widely used as a means of invalidating the local deterministic theories and a tool of device-independent quantum cryptographic tasks. There exists a randomness (free will) loophole in the test, which is widely believed impossible to be closed perfectly, that is, certain random inputs are required for the test. Following a randomness quantification method used in literature, we investigate the randomness required in the test under various assumptions. By comparing the results, we conclude that, in order to make the test result reliable, it is more important to rule out the correlation between multiple runs than the correlation between two parties.

Figure 1

Bell tests in the bipartite scenario. (a) The inputs of Alice and Bob, $x$ and $y$, are decided by perfect random number generators (RNGs), which produce uniformly distributed random numbers. (b) The measurement devices are controlled by an adversary Eve through local hidden variables $\lambda$. (c) The input random numbers are also controlled by the same local hidden variable $\lambda$, which is accessible to Eve.

Figure 2

Optimal values of the CHSH test for different randomness $P$ with various rounds $N$ based on only Alice's inputs biased when conditioned on the hidden variable $\lambda$. The solid line is the optimal strategy for $N\to \infty$, which upper bounds all finite $N$ rounds. Note that the curve is not smooth for finite runs $N$ because the optimal strategy $q_{k_A}$ defined in Eq. (27) jumps on $l$. With $N$ grows larger, the curve tends to be smoother.

Figure 3

Possible optimal values of the CHSH test for different randomness $P$ with various rounds $N$ based on uncorrelated inputs of Alice and Bob. The solid line corresponds the strategy for $N\to \infty$, which upper bounds all finite $N$ cases. The curves are not smooth for finite $N$ for similar reasons like in the one-party-biased case, and it tends to be smooth with $N\to \infty$.