Bell derives the inequality $$|E(\vec{a},\vec{b})-E(\vec{a},\vec{c})|\leq 1+E(\vec{b},\vec{c})$$ in his book Speakable and unspeakable in quantum mechanics. In this derivation he uses the assumption that when the axes $\vec{a}$, $\vec{b}$ are aligned, the outcomes of measurements $A,B$ of the spins of spacelike-seperated particles 1,2 along these axes respectively are anticorrelated so that the product $AB$=-1.

Other similar "Bell-type" inequalities such as the CHSH inequality do not use this assumption. The CHSH inequality has the form

$$-2\leq E(\vec{a},\vec{b}) + E(\vec{a},\vec{b}') - E(\vec{a}',\vec{b}) + E(\vec{a}',\vec{b}')\leq 2 $$

My question is, is there anything other than the above assumption made in the derivation of Bell's original inequality that makes it unsuitable to experimental testing? I.e. is it something to do with the form of Bell's inequality?

Most papers I've read just simply state that the original inequality isn't suitable for testing and gloss over some statement about perfect correlations.


1 Answer 1


The original derivation assumed that every measurement would give a result, such as for example "particle with up-spin detected," or "particle with down-spin detected" But in real experiments some particles never affected the detectors; presumably they "leaked out" or somehow vanished, and no measurements occurred. Hence subsequent derivations provide for non-detections, usually scored as a zero result. E.g. the CHSH derivation considers three possible scores, +1, -1 and 0. This is why the S value that the inequality is solved for (if local hidden variables exist) calls for "S equal to or less than |2|," rather than just "S less than |2|." In other words the inequality is satisfied if S falls anywhere between -2 and +2. Of course non-detections are serious problems because they allow the fair-sampling loophole, but that's another matter: If 1000 entangled particles were launched and only 100 were measured, are these 100 representative of the entire population of entangled particles? This question could not even be considered by the original derivations.


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