Abstract
Bell's theorem is a no-go theorem stating that quantum mechanics cannot be reproduced by a physical theory based on realism, freedom to choose experimental settings and two locality conditions: setting (SI) and outcome (OI) independence. We provide a novel analysis of what it takes to violate Bell's inequality within the framework in which both realism and freedom of choice are assumed, by showing that it is impossible to model a violation without having information in one laboratory about both the setting and the outcome at the distant one. While it is possible that outcome information can be revealed from shared hidden variables, the assumed experimenter's freedom to choose the settings ensures that the setting information must be non-locally transferred even when the SI condition is obeyed. The amount of transmitted information about the setting that is sufficient to violate the CHSH inequality up to its quantum mechanical maximum is 0.736 bits.
Highlights
Bell’s inequalities are certain constraints on correlations between space-like separated measurements which are satisfied in any local realistic theory [1]
Our analysis shows that SI and outcome independence (OI) do not provide us with the full picture of what local information is needed in violations of Bell inequalities
If there is no setting information, the violation of Ineq. (4), or Ineq. (1), is impossible. Both the information about the distant setting and about the distant outcome must be available at the local laboratory to have a violation, we show that, given freedom of choice, the information about the distant setting has to be transmitted non-locally, whereas it is possible that the information about the distant outcome can be obtained without any transmission from the shared hidden variables
Summary
Which is sufficient for our analysis and to which we refer as freedom of choice throughout the rest of the paper. Both the information about the distant setting and about the distant outcome must be available at the local laboratory to have a violation, we show that, given freedom of choice, the information about the distant setting has to be transmitted non-locally, whereas it is possible that the information about the distant outcome can be obtained without any transmission from the shared hidden variables. This allows us to analyze the above mentioned asymmetry between the outcome and setting information in a more formal way To this end, we introduce a measure of information, that we call “transmitted ‘information’” (TI), which is the difference of the averaged probability of correctly guessing the value of the variable Y when knowing X and λ, and the one when knowing only λ:. The asymmetry between the outcome and setting information originates from the freedom of choice assumption
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