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Clauser-Horne Bell test with imperfect random inputs

Bell test is one of the most important tools in quantum information science. On the one hand, it enables fundamental test for the physics laws of nature, and on the other hand, it can be also applied in varieties 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. With partially random inputs, we explicitly bound the Bell value for all local hidden variable models by optimizing the classical strategy. Our result thus puts input randomness requirement on the Clauser-Horne test under varieties of practical scenarios. The employed analysis technique can be generalized to other Bell's inequalities.