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No-signaling condition and RI
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Relativistic independence bounds nonlocality

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A consequence of relativistic causality in probabilistic theories is the no-signaling condition (1). Consider the Bell-CHSH setting where a and b are the outcomes of Alice’s and Bob’s measurements. The joint probability of these outcomes when Alice measured using device i and Bob measured using device j is denoted as p(a, b | i, j). The no-signaling condition states that one experimenter’s marginal probabilities are independent of another experimenter’s choices, namely$∑bp(a,b|i,0)=∑bp(a,b|i,1)def¯¯p(a|i)∑ap(a,b|0,j)=∑ap(a,b|1,j)def¯¯p(b|j)$(7)

Of course it means that one experimenter’s precision is independent of another experimenter’s choices$ΔAi2=Ea2|i,j−Ea|i,j2=∑a,ba2p(a,b|i,j)−(∑a,bap(a,b|i,j))2=∑aa2p(a|i)−(∑aap(a|i))2ΔBj2=Eb2|i,j−Eb|i,j2=∑a,bb2p(a,b|i,j)−(∑a,bbp(a,b|i,j))2=∑bb2p(b|j)−(∑bbp(b|j))2$(8)

The no-signaling condition thus implies that the variances of one experimenter in the Alice-Bob uncertainty relations (Eq. 3) are independent of the other experimenter’s choices.

RI implies that one experimenter’s uncertainty relation is altogether independent of the other experimenter’s choices, i.e., that ΛA as a whole, and therefore also rj, are independent of j. This does not necessarily imply the no-signaling condition, as there may exist, for example, marginal distributions p(a | i, j) that depend on Bob’s j whose variances, $ΔAi2$, are nevertheless independent of this j. This shows that RI does not at all require us to assume the no-signaling condition.

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