I am under the impression that violations of Bell's inequality as shown in e.g. the Aspect experiment can be explained by the fact that the particles where not separable rather then the non-existence of local hidden variables. But what does separable mean in this context?
It seems to me that the particles are already 'not separable' due to them been in an entangled state and as such we are not in a position to deny local hidden variables.
The notion of separability of a state has a precise and simple meaning:
In natural language, a separable state of a system that has several subsystems is a state to which a unique state of every subsystem is associated.
In classical mechanics, all states are separable in this sense - given two configuration spaces $Q_1,Q_2$, the configuration space of the combined system is the direct product $Q_1\times Q_2$, which, by definition, just consists of pairs of states $(q_1,q_2)$ with $q_1\in Q_1$ and $q_2 \in Q_2$.
In quantum mechanics, the entangled states are by definition those which are not separable. Given two Hilbert spaces of state $H_1,H_2$, the combined space of state is the tensor product $H_1\otimes H_2$, which is spanned by simple tensors of the form $\psi \otimes \phi, \psi \in H_1, \phi\in H_2$, but the sum of simple tensors (which are the separable states) is not necessarily a simple tensor. Any state $\chi \in H_1\otimes H_2$ which cannot be written as $\psi\otimes\phi$ for some states of the subsystems, but only as a sum, is called non-separable or entangled - precisely because there are no unique substates, so we cannot "separate" the state into its subsystems.
The word separable (the property) has a precise and detailed meaning (about lack of factorizability) when discussing multiparticle states. But that meaning might not be the meaning in the context you consider.
For instance the paper http://arxiv.org/abs/1302.7188 argues that separability (the principle) is not related to Bell's inequality. And in that paper separability is a principle, like locality, that you can use (or not) to make Bell's nequality. So one person might argue that you need hidden variables, locality, and separability to get an inequality (which I suspect is what your unnamed source claims) whereas the paper I link (thanks to Luboš Motl for mentioning the paper) might argue that only hidden variables and locality are required.
OK, so what is separability (the principle) in the context of whether it is an assumption? Even more unfortunately, different people have used different ones, Howard has one, Henson has another. In Henson's is an assumption about a kind of compositionality of sigma algebras associated with regions where the sigma algebra associated with a (countable) union of disjoint regions is exactly equal to the sigma algebra generated by the union of the sigma algebras associated with the disjoint parts.
This principle can never be used to infer anything at all, so it doesn't have to be used to establish Bell's inequality or to establish anything ever, it is a principle devoid of content. This might sound overly harsh, but it is true. You can try jumping straight to the paragraph beginning "For instance start with any separable theory that has a region $P$ with at least one point $p$ in it where $\dots$" to see the argument why, basically there are always a nonseparable thing that acts just like any given separable thing so you never "need" to be separable and never "have" to assume it.
The final paragraph explains why, based on Bells original paper the separability property (non-entangled)/is used for Bell, whereas this separability principle I will now show is now used by Bell. But unlike Henson I'll just show that it is pointless for anything.
Henson's paper (the arxiv version) is unclear on the one hand about why anyone would ever want to consider a separable system of sigma algebras associated with regions or on the other hand how you could even consider a nonseparable space in any useful way that is consistent with how we normally look at experimental outcomes as results of stuff that is made and calibrated and operated out of smaller stuff (which would seem to make it separable in a natural way).
The paper (the arxiv version) has an example of a local nonseparable space by fiat that simply has an event that appears in the algebra associated with a region that has no generators in any of a disjoint partition of it (well I added the partition part because otherwise the explicit example might not be nonseparable, so clearly what the author must have meant). Later I will use the exact same trick to show the whole idea is pointless.
The paper (the arxiv version) sometimes reads as confused in ways that cannot be fixed so simply as when i added the word partition for instance when it brings up nonsimply connected spaces in a context where neither paths nor contractibility nor even topology at all is considered, so just utter nonsense unrelated to anything in the article and hopefully not put there to intimidate readers. So it is possible that the definition of Separability could be changed yet again in another paper.
Back to how I mentioned it was unclear how to even have nonseparable algebras lime the ones we use? The sample space is defined in such a vague way as to give an almost tautological positivist kind of sample space, just as a set of histories, so any event (in the probabilist meaning, not the relativity meaning) is basically the set of histories consistent with an event. For instance if a Stern-Gerlach device reads spin up, the set of histories consistent with that is the event. As far as I know this is not intended to be related Robert Griffith's consistent histories, the quantum interpretation. Now back to the Stern-Gerlach, if it has parts such as a place for beams to go in and some screens for deflected beams to register hits and a place for the magnet for the inhomogeneous field and a some shielding to prevent stay hits by other things and so on, then you can characterize how the device functions by the sum if its parts, such as screen registers electron, shielding doesn't get destroyed, magnet stays in place, beam enters device, etc. Each of those is presumably an event and, so therefore is their intersection, which is the event of the Stern-Gerlach registering an event. To have a nonseparable algebra you need an event that can't be decomposed in terms of the workings and functions of its parts or an event (in the probabilist's sense of the word) that can't be described in terms of how its spatially separated parts work. And the fact that such vagueness is intended to be covered by the definition is demonstrated by the author of the paper when only later does the author chooses to make events be ... about things that Physics normally concerns itself. So before then it could have included any possible criteria even untestable ones.
Having an example by fiat that basically just says let's not describe how those smaller parts worked and wow now the algebra is mysteriously nonseparable, it doesn't say much. If anything it tells you that you have too much power and freedom to choose which things to call events. If choosing to call something an event or not makes it non separable how physical can it be?
To be fair, maybe the point is whether there are quantum events that can't be described as localized events or can't be described in terms of smaller parts, so I could be too harsh. But the point shouldn't simply whether you defined a term called separability that may or may not apply to things (though if it couldn't then its utility to science is rather limited) but whether we specifically allow separable and nonseparable theories when proving Bell's inequality.
When Henson does their proof they formulated a special kind of locality called Bell Locality that they then uses instead of using Locality or Separability. And then concludes that they didn't use Separability. But unlike in the comparison between Locality and Separability where they it was shown to be possible to have Locality with Separability by utilizing the freedom to declare something to not be an event. This time events are required to include all those things that Physics normally concerns itself, including operationally defined outcomes and setting of devices (which are then contained in the algebra for that region if they can be done in that region). And so the author does not explicitly say that you can be non separable while still having Bell Locality. But you can. However the ways I can show it can be done are silly.
For instance start with any separable theory that has a region $P$ with at least one point $p$ in it where the region has a sigma algebra associated with the region that is not the trivial sigma algebra of $\{\emptyset, \overline\emptyset\}.$ For instance the point could be the center of the incoming beam of a Stern-Gerlach device at a fixed time (I'm saying point in spacetime since the word event is being used in the probabilist's sense). You need that point $p$ to do measurements then and there so call that measurement an event E that is not $\emptyset$ or $\overline\emptyset.$ In other words $\emptyset\neq E\neq\overline\emptyset.$ Now fix that point $ p.$
Then make a whole new spacetime that looks like the last one except instead of one point at that point $p$ it has two, say a red point r and a green point g. And for every region B in the original spacetime make a region B' that is just like it except that if the original region contained p the corresponding one contains both r and g and for each region of the new spacetime have the associated sigma algebra of observables be exactly the same for the corresponding region. This means we have two things that currently are exactly the same they have the same events and they make the exact same predictions about the probability of every event including correlations and conditional probabilities.
Alright. We have two things that can't be experimentally distinguished. We know what Occam would prefer, which is the one with p instead of r and g, but they make the same predictions so it isn't a big deal. Now we will change our new thing (the one with r and g though right now they always do the same things). We are going to change our new thing to make it not separable while not changing a single testable prediction of the theory. We do this by making new regions and associated sigma algebras that introduce no new events and change no probabilities but merely make our new thing not separable. Alright. To do this we are going to add two new regions, $\{a\in P : a \neq p\}\cup\{r\}$ and $\{g\}$ and have the sigma algebra associated with each be the trivial algebra $\{emptyset,\overline\emptyset\}.$ Now the disjoint union of these two regions is P' and so its associated sigma algebra is the same as P and so contains E but the two trivial sigma algebras do not generate an algebra that contains E, so our new thing is not separable even though it makes the same testable prediction (same events, same probabilities, same correlations, same conditional probabilities).
So we don't have to use the Principle of Separability because whenever you have a setup that satisfies the principle of Separability, you can trivially make one that doesn't that makes all the same predictions about what you see in the lab (we have the same events that can happen in a representation of the original sigma algebras of an associated regions with the same probabilities, just we have some extra small regions that have no purpose expect to fail to have events in them, in particular that allow no new experiments and no new science).
It's a silly definition, separability (the principle) doesn't imply anything scientific. So you never use it for anything. But I think I've seen this question asked before so I thought I should answer it.
Finally, why do I say the other definition is not the one you are talking about? Bell explicitly assumed an entangled state in equation 13 of his original paper so there should be no debate about that use of the word (separable, the property, as a synonym for non entangled) unless maybe you consider potentially different proofs of the same result.
