The meaning of “violation of Bell’s inequalities”
I can safely say that I have known for long time what is the meaning of Bell’s inequalities, and I have even tried to convey it (I’ll post my usual presentation soon, for comparison). But yesterday, I heard one of the clearest expositions ever, which may even re-shape the way I’ll present these topics in the future. It was a talk by Michael Hall, based on one of his recent works. Let me try to summarize it here, for everyone’s benefit and my own record.
The starting point is an attempt at explaining quantum correlations as deriving from some “underlying information” [a much better name than “local hidden variable” indeed!]. In this context, one would like to make three very plausible assumptions:
- Determinism (or equivalently “outcome independence”): the “underlying information” specifies the outcomes (or: given the “underlying information”, there are no correlations).
- No-signaling: even knowing the “underlying information”, the marginal statistics of one particle are independent of the measurement performed on the other particle.
- Measurement independence: the “underlying information” does not have any influence on the measurement settings that are used.
Given these assumptions, Bell’s inequalities are obtained. Since quantum correlations violate the inequalities, at least one of the three assumptions must be wrong.
In the standard interpretation of QM, the “underlying information” is the quantum state; 1 is fully denied, while 2 and 3 are retained. Personally, I stand clearly here.
In Bohmiam mechanics, the “underlying information” is the quantum potential, which must change instantaneously everywhere upon local measurement: therefore, 2 is denied, while 1 and 3 are retained.
It seems harder to give up 3: it would mean that the choice of the setting by (ideally) a human being would be influenced by the source. Or is it really harder? Many physicists I talked to, once sufficiently pressed, admit advocating a fully deterministic view of the world, in which free will is an illusion (I find it terrible, but as we know, there is no way of disproving someone who believes this).
Anyway, as Michael said, this is “philosophy” (albeit interesting one). What he did, was to try to quantify: “how much” determinism, no-signaling or free will must one give up, in order to reproduce the observations? His preliminary studies indicate that:
- if you want to give up determinism, you have to give it up fully, i.e. one bit per pair of particles (an extension of something we proved back in 2008);
- if you want to give up no-signaling, you have to signal quite a lot too.
- However, it is enough to give up very little of free will: only 1/15 of a bit per pair of particles, according to some figure of merit.
As for myself, I am not ready to give up even that amount of free will; but it’s intriguing nonetheless.
A last note: in the discussion, Rafael Rabelo pointed out another assumption in the derivation of Bell’s theorem: the fact that the devices have no memory, i.e. they do not take into account the settings and outcomes of the previous runs. I seem to remember that this changes little, but who knows? Anyway, see how a competent and aware student may remind an expert of something that his “fully general” analysis did not take into account 🙂