No action at a distance, spooky or not

 On the occasion of the announcement of the Nobel prize for Aspect, Clauser and Zeilinger for the experimental verification that quantum theory violates Bell's inequality, there seems to be a strong urge in popular explanations to state that this proves that quantum theory is non-local, that entanglement is somehow a strong bond between quantum systems and people quote Einstein on the "spooky action at a distance".

But it should be clear (and I have talked about this here before) that this is not a necessary consequence of the Bell inequality violation. There is a way to keep locality in quantum theory (at the price of "realism" in a technical sense as we will see below). And that is not just a convenience: In fact, quantum field theory (and the whole idea of a field mediating interactions between distant entities like the earth and the sun) is built on the idea of locality. This is most strongly emphasised in the Haag-Kastler approach (algebraic quantum field theory), where pretty much everything is encoded in the algebras of observables that can be measured in local regions and how these algebras fit into each other. So throwing out locality with the bath water removes the basis of QFT. And I am convinced this is the origin why there is no good version of QFT in the Bohmian approach (which famously sacrifices locality to preserve realism, something some of the proponents not even acknowledge as an assumption as it is there in the classical theory and it needs some abstraction to realise it is actually an assumption and not god given).

But let's get technical. To be specific, I will use the CHSH version of the Bell inequality (but you could as well use the original one or the GHZ version as Coleman does). This is about particles that have two different properties, here termed A and B. These can be measured and the outcome of this measurement can be either +1 or -1. An example could be spin 1/2 particles and A and B representing twice the components of the spin in either the x or the y direction respectively.

Now, we have two such particles with these properties A and B for particle 1 and A' and B' for particle 2. CHSH instruct you to look at the expectation value of the combined observable

\[A (A'+B') + B (A'-B').\]

Let's first do the classical analysis: We don't know about the two properties of particle 2, in the primed variables. But we know, they are either equal or different. In case they are equal, the absolute value of A'+B' is 2 while A'-B'=0. If they are different, we have A'+B'=0 while the absolute value of A'-B' is two. In either case, one one of the two terms contribute and in absolute value it is 2 times the unprimed observable of particle one, A for equal values in particle 2 an B for different values for particle 2. No matter which possibility is realised, the absolute value of this observable is always 2.

If you allow for probabilistic outcomes of the measurements, you can convince yourself that you can also realise smaller absolute values than 2 but never larger ones. So much for the classical analysis.

In quantum theory, you can, however, write down an entangled state of the two particle system (in the spin 1/2 case specifically) where this expectation value is 2 times the square root of 2, so larger than all the possible classical values. But didn't we just prove it cannot be larger than 2?

If you are ready to give up locality you can now say that there is a non-local interaction that tells particle 2 if we measure A or B on particle one and by this adjust its value that is measured at the site of particle two. This is, I presume, what the Bohmians would argue (even though I have never seen a version of this experiment spelled out in detail in the Bohmian setting with a full analysis of the particles following the guiding equation).

But as I said above, I would rather give up realism: In the formula above and the classical argument, we say things like "A' and B' are either the same or opposite". Note, however, that in the case of spins, you cannot both measure the spin in x and in y direction on the same particle because they do not commute and there is the uncertainty relation. You can measure either of them but once you decided you cannot measure the other (in the same round of the experiment). To give up realism simply means that you don't try to assign a value to an observable that you cannot measure because it is not compatible with what you actually measure. If you measure the spin in x direction it is no longer the case that the spin in the y direction is either +1/2 or -1/2 and you just don't know because you did not measure it, in the non-realistic theory you must not assign any value to it if you measured the x spin. (Of course you can still measure A+B, but that is a spin in a diagonal direction and then you don't measure either the x nor the y spin).

You just have to refuse to make statements like "the spin in x and y directions are either the same or opposite" as they involve things that cannot all be measured, so this statement would be non-observable anyways. And without these types of statement, the "proof" of the inequality goes down the drain and this is how the quantum theory can avoid it. Just don't talk about things you cannot measure in principle (metaphysical statements if you like) and you can keep our beloved locality.