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The correlations that violate the CHSH inequality are known to havecomplementary contributions from signaling and local indeterminacy. Thiscomplementarity is shown to represent a strengthening of Bell's theorem, andcan be used to certify randomness in a device-independent way, assuming neitherthe validity of quantum mechanics nor even no-signaling. We obtain generalnonlocal resources that can simulate the statistics of the singlet state,encompassing existing results. We prove a conjecture due to Hall (2010) and Karet al. (2011) on the complementarity for such resources.

Complementarity between signalling and local indeterminacy in quantum nonlocal correlations