I'm studying Bell's theorem and the CHSH inequality for some time. Now it's clear to me that one cannot reproduce the correlations predicted by quantum mechanics by assuming that particles carry hidden variables with them, and measurements depend only on them.

But what about the case where we consider the most general scenario: the measurement outcome can depend on the entire past light cone. Why not?

This would mean more information are available when the detector "chooses" an outcome, than just the variables carried by the current particle: for example the measurement of the current particle can depend on previous measurements, past measurements on the other detector when they reached the past light cone of the current event, etc.

Is it possible to show that such assumptions are also ruled out Bell's theorem?

  • $\begingroup$ Doesn't the local in the name of the ruled out local hidden variable theories exactly mean that hidden variables only help if they are non-local, i.e. influences can come from outside the light-cone? $\endgroup$ – ACuriousMind Feb 11 '15 at 18:25
  • $\begingroup$ @Calmarius I support ACuriousMind's answer, just I'd like to make it more explicit. Assume a measurement on two entangled particles, one coming to Alice's lab and one coming to Bob's lab. And assume that each lab is installed on another space-station. And also assume that the Alice and Bob measure at the same time by the Earth clock. From the point of view of a traveler moving toward Alice's station, Alice measures first, and Bob later. But from the point of view of a traveler moving toward Bob's station, Bob measures first, and Alice later. But the results are correlated. (I continue) $\endgroup$ – Sofia Feb 11 '15 at 20:39
  • $\begingroup$ @Calmarius : then, is the result obtained by someone of the experimenters, dependent only on the past light-cone of his/her measurement, be it the entire light cone? Answer : entanglements work outside the space-time. Time, light-cone mean nothing to them. $\endgroup$ – Sofia Feb 11 '15 at 20:43
  • $\begingroup$ All such influences are already encoded into the idea of "local hidden variables" - for instance imagine every point keeps a (hidden) memory of every event in its past light cone. Those data are still local hidden variables. $\endgroup$ – adipy Feb 11 '15 at 21:24

Yes, as long as you're assuming a local hidden variables theory it can be shown that even allowing the outcome to be determined by any arbitrary amount of prior events in the past light cone will not allow for violations of Bell inequalities. Bell demonstrates this for example in his paper "La nouvelle cuisine" which is reprinted in the collection Speakable and Unspeakable in Quantum Mechanics. For a free online paper that discusses how you can include entire cross-sections of the past light cone in proofs of Bell's theorem, see "J.S. Bell's Concept of Local Causality"--note in particular Fig. 1 and Fig. 2 on page 4 of the paper, and the way equation (1) on that page defines the locality condition using the complete set of "beables" (all local variables, whether measurable or hidden variables) $B_3$ in a cross-section of the past light cone (region 3 in Fig. 2).

  • $\begingroup$ to be clear, if you allow the measurement choices to also correlate with the hidden variable, then Bell's theorem does not hold. In other words, in some sense, allowing correlations with arbitrary events in the past of the measurement outcomes does allow to violate Bell's inequalities (trivially, because in such a scenario there are no restrictions on the output probability distributions) $\endgroup$ – glS Jul 19 '18 at 10:59

Yes, Bell's theorem (together with the Einstein-Podolsky-Rosen argument) necessarily implies that causality is nonlocal, i.e. causal connections outside the past light cone exist. So the past light cone is not sufficient to determine all measurements.

Note that this is regardless of whether hidden variables exist or not. This is an often-misunderstood point. It's not, "Choose your poison: either hidden variables exist and causality is nonlocal, or hidden variables do not exist and causality is local." It's, "Causality is nonlocal. Period."

(Note, BTW, that your statement in the first paragraph is not true: you CAN reproduce QM correlations with a hidden variable theory, but that theory will be nonlocal. David Bohm invented pilot wave theory to demonstrate that a hidden variable theory is capable of reproducing QM correlations.)

The logic is like this: EPR says, in effect, "If QM is true, and causality is local, then hidden variables exist." Bell's theorem says, "If hidden variables exist, and causality is local, then QM is false." If the experiments demonstrate "QM is true" (as most people think they do), then those syllogisms become: 1.(EPR) If causality is local, then hidden variables exist. 2. (Bell) Either hidden variables do not exist, or causality is nonlocal. Combining those: if causality is local, then (by EPR) hidden variables exist, so (by Bell) causality is nonlocal (contradiction).

  • $\begingroup$ Violations of Bell's theorem don't necessarily imply "causality is nonlocal", they just imply that the type of theories defined as "local realist" theories cannot be correct. Local realism assumes for example that each measurement yields a unique result, so if you violate that condition, you can get a theory that isn't "local realist" but features no nonlocality. $\endgroup$ – Hypnosifl Mar 3 '15 at 3:47
  • $\begingroup$ @Hypnosifl: Can you explain the difference between a local hidden variables theory and a local realistic theory? $\endgroup$ – pwf Mar 3 '15 at 6:15
  • $\begingroup$ A local hidden variables theory is just a type of local realist model, one where there are local physical variables beyond the ones that can be determined from the quantum state of a system. If quantum mechanics is correct as far as predictions of measurement results, this would imply that even if such extra variables exist they would be impossible for us to measure, hence the name "hidden". $\endgroup$ – Hypnosifl Mar 3 '15 at 11:21
  • $\begingroup$ @Hypnosifl: Bell's theorem says any local theory that is consistent with QM must be unrealistic. EPR says any local theory that is consistent with QM must be realistic (at least w.r.t. the relevant conjugate variables). Ergo, if the QM correlations are right, local theories are out, one way or the other. $\endgroup$ – pwf Mar 3 '15 at 17:09
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    $\begingroup$ Is that your own stab at a definition, or have you seen any physicists define "realistic" this way? If "realistic" simply meant "deterministic" as you seem to suggest, then why wouldn't they use the phrase "local deterministic"? Also, if I'm not mistaken Bell's theorem is general enough to show that Bell inequalities shouldn't be violated in a local but fundamentally stochastic theory, it doesn't depend on any assumption of determinism. $\endgroup$ – Hypnosifl Mar 3 '15 at 21:14

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