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arXiv:1006.3680 · Physical Review A

Complementary contributions of indeterminism and signaling to quantum correlations

Michael J. W. Hall

Published Dec 28, 2010• 32 citations
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Abstract

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 $1\ensuremath{-}p$.

DOI
10.1103/physreva.82.062117

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Simple quantitative measures of indeterminism and signaling, III and SSS, 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)B(I,S)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 8080%80 probability of occurrence to some underlying event, and/or allow a nonlocal change of at least 6060%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 ppp to its complementary value 1\ensuremath−p1\ensuremath{-}p1\ensuremath−p.
The correlations that violate the CHSH inequality are known to have complementary contributions from signaling and local indeterminacy. This complementarity is shown to represent a strengthening of Bell’s theorem, and can be used to certify randomness in a device-independent way, assuming neither the validity of quantum mechanics nor even no-signaling. We obtain general nonlocal resources that can simulate the statistics of the singlet state, encompassing existing results. We prove a conjecture due to Hall (2010) and Kar et al. (2011) on the complementarity for such resources. Further, we show that signaling leads to biasing of the marginals for these resources. This entails that no-signaling is a consequence of isotropic unbiasedness of the singlet statistics, by which is meant that the marginals are unbiased along any direction, and coincidence counts are determined only by the relative orientation of pairs of polarizers.
The correlations that violate the CHSH inequality are known to have complementary contributions from signaling and local indeterminacy. This complementarity is shown to represent a strengthening of Bell's theorem, and can be used to certify randomness in a device-independent way, assuming neither the validity of quantum mechanics nor even no-signaling. We obtain general nonlocal resources that can simulate the statistics of the singlet state, encompassing existing results. We prove a conjecture due to Hall (2010) and Kar et al. (2011) on the complementarity for such resources.
The derivation of Bell inequalities requires an assumption of measurement independence, related to the amount of free will experimenters have in choosing measurement settings. Violation of these inequalities by singlet state correlations brings this assumption into question. A simple measure of the degree of measurement independence is defined for correlation models, and it is shown that all spin correlations of a singlet state can be modeled via giving up just 14% of measurement independence. The underlying model is deterministic and no signaling. It may thus be favorably compared with other underlying models of the singlet state, which require maximum indeterminism or maximum signaling. A local deterministic model is also given that achieves the maximum possible violation of the well-known Bell-Clauser-Horne-Shimony-Holt inequality, at a cost of only 1/3 of measurement independence.
The correlations that violate the CHSH inequality are known to have complementary contributions from signaling and local indeterminacy. This complementarity is shown to represent a strengthening of Bell's theorem, and can be used to certify randomness in a device-independent way, assuming neither the validity of quantum mechanics nor even no-signaling. We obtain general nonlocal resources that can simulate the statistics of the singlet state, encompassing existing results. We prove a conjecture due to Hall (2010) and Kar et al. (2011) on the complementarity for such resources.
Violation of the Bell inequality (or Bell-type inequalities) by nonlocal correlations is justified by relaxation of at least one of the plausible physical constraints used to model such inequality. Based on this fact, in this Brief Report we present a procedure to simulate three-qubit Greenberger-Horne-Zeilinger correlation relaxing two constraints---determinism and no signaling---simultaneously. We have also derived the minimum amount of indeterminism and signaling to be introduced in a system. The corresponding number of signaling and local bits of mutual information needed to communicate are also provided, and thus we are able to focus on the utility of relaxation of these two constraints as useful resources.
Quantum correlations that violate the Bell inequality cannot be explained by any local-hidden-variable theory that is measurement independent. However, this violation merely signifies the incompatibility of the underlying assumptions of reality, locality, and measurement independence, without providing a quantitative measure of the extent to which each assumption is violated. In contrast, Hall introduced a measure for each of the following assumptions: indeterminism, signaling, and measurement dependence, and Hall also generalized the Bell-Clauser-Horne-Shimony-Holt inequality, which provides a quantitative trade-off relationship between these assumptions. In this paper, we consider the introduction of hidden variables to be an essential assumption of Bell's theorem and introduce a quantification of hidden variables, which we term ``hiddenness.'' We derive a trade-off relation between hiddenness and measurement dependence that applies to any local-hidden-variable theory.