Clauser-Horne Bell test with imperfect random inputs

The Bell test is one of the most important tools in quantum information science. On the one hand, it enables fundamental tests of the basic physics laws of nature, and on the other hand, it can also be applied in a variety of device-independent tasks such as quantum key distribution and random number generation. In practice, loopholes existing in experimental demonstrations of Bell tests may affect the validity of the conclusions. In this work, we focus on the randomness (freewill) loophole and investigate the randomness requirement in a well-known Bell test, the Clauser-Horne test, under various conditions. We explicitly bound the Bell value for all local hidden variable models by optimizing over all classical strategies exploiting the knowledge of the partially random inputs. Our result thus provides input randomness requirements on the Clauser-Horne test under varieties of practical scenarios. The employed analysis technique can also be generalized to other Bell inequalities.

Figure 1

Bell tests in a bipartite scenario. In general, the inputs depend on some local hidden variable $\lambda$. The local hidden variables that control the inputs and the devices may be different. However, we can still represent these two local hidden variables as a single one denoted $\lambda$.

Figure 2

CH value $J_{\mathrm{CH}}^{\mathrm{LHVM}}$ as a function of $P$ and $Q$, according to Eq. (18).

Figure 3

CH value $J_{\mathrm{CH}}^{\mathrm{LHVM},\mathrm{Fac}}$ as a function of $P$ and $Q$ with the factorizable condition, Eq. (7).

Figure 4

CH value $J_{\mathrm{CH}}^{\mathrm{LHVM},\mathrm{NS}}$ as a function of $P$ and $Q$ under the NS condition, Eq. (22).

Figure 5

CH value $J_{\mathrm{CH}}^{\mathrm{LHVM},\mathrm{NS},\mathrm{Fac}}$ as a function of $P$ and $Q$ under factorizable [Eq. (7)] and NS [Eq. (22)] conditions.

Figure 6

Critical value of $Q$ and $P$ such that the CH value $J_{\mathrm{CH}}^{\mathrm{LHVM}}(P,Q)$ equals the maximal quantum value $J_Q=(\sqrt{2}-1)/2$.

Figure 7

CH value $J_{\mathrm{CH}}^{\mathrm{LHVM}}\left(\delta \right)$ under different conditions.