Simple quantitative measures of indeterminism and signaling, $I$ and $S$, are defined for models of statistical correlations. It is shown that any such model satisfies a generalized Bell-type inequality, with tight upper bound $B(I,S)$. This upper bound explicitly quantifies the complementary contributions required from indeterminism and signaling, for modeling any given violation of the standard Bell–Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. For example, all models of the maximum quantum violation must either assign no more than $80\%$ probability of occurrence to some underlying event, and/or allow a nonlocal change of at least $60\%$ in an underlying marginal probability of one observer in response to a change in measurement setting by a distant observer. The results yield a corresponding complementarity relation between the numbers of local random bits and nonlocal signaling bits required to model a given violation. A stronger relation is conjectured for simulations of singlet states. Signaling appears to be a useful resource only if a “gap” condition is satisfied, corresponding to being able to nonlocally flip some underlying marginal probability $p$ to its complementary value $1-p$.

• Received 18 June 2010

DOI:https://doi.org/10.1103/PhysRevA.82.062117