# Bell's theorem outside of the context of entanglement

I am an undergrad college student and have been watching videos on YouTube explaining Bell's theorem / inequality and how it shows that there cannot be local hidden-variable theories for entanglement.

However, the experiments most videos talk about are in the context of entanglement specifically, where the experiment involves taking a pair of entangled particles and recording the statistics of how often the results match when two people repeatedly measure the two particles along randomly chosen axes. The statistics of the data we get after performing the experiment rules out local hidden variables.

My question is how this rules out local hidden variable theories in quantum mechanics outside of the context of entanglement? Or does it actually only rule out local hidden variable theories in the context of entanglement?

For example, we could have a local hidden variable theory outside of entanglement like the following: A single electron (no entanglement) could have a definite position before it is measured (a local hidden variable), but we cannot know it until we have measured it, and that local hidden variable is why we can only talk about the probability of finding it at a location. And therefore, the randomness is not a fundamental property of the quantum nature of the electron but is due to a hidden variable.

It seems to me that the Bell's theorem experiment they talk about in the videos only tells us that quantum entanglement specifically cannot be explained by local hidden-variable theories, but not that other quantum phenomena, such as the fact that we can only know the probability of finding an electron in an atom at a certain location until we measure it, cannot be explained by local hidden-variable theories.

Are there other experiments that have been done that rule out local hidden variable theories in these other contexts? Or if not, how do you generalize the result of Bell's theorem/experiment in the "entanglement case" to prove there cannot be local hidden-variable theories in quantum mechanics in general?

Clarification: I am not for or against using any theory. I just don't see how local hidden variable theories in QM in general, are ruled out by the Bell experiments with entanglement specifically and am asking how/if local hidden variable theories are ruled out in general.

• To be clear, you are proposing that we use a local hidden variable theory (whose predictions match that of QM) in those situations where we can get away with it, and resort to quantum mechanics whenever it can be proved that no local hidden variable theory could match the (verifiably correct) predictions of QM, right? Why wouldn't we just use QM all the time? Commented Aug 19, 2022 at 22:57
• I completely agree with @J.Murray . But for the record, yes, you are right that entangled states are precisely the states where there are no joint probability distributions for the outcomes of various observations, and so they are precisely the states where no hidden variable story can work. Commented Aug 19, 2022 at 23:11
• Bell’s theorem says that QM can produce correlations that cannot be produced by a local hidden variable theory. It doesn’t say QM must produce such correlations in all cases.
– d_b
Commented Aug 19, 2022 at 23:32
• you don't need two particles to violate a Bell inequality. You just need different degrees of freedom. You can violate a Bell inequality using an "entangled state" of, say, polarization and position degrees of freedom of a single photon/particle
– glS
Commented Aug 22, 2022 at 14:54
• For example, how do you explain the single photon mach zehnder experiment if your image of a photon or electron or whatever is as a single distinct ball with specific (but unknown) properties at any given moment?
– TKoL
Commented Aug 23, 2022 at 16:59

## 1 Answer

My question is how does this rule out local hidden variable theories in quantum mechanics outside of the context of entanglement?

It doesn't! That is a good point, Bell's theorem applies exactly to the system that Bell described and it does not globally rule out the existence of hidden variables. And, though there are many other systems for which you can come up with similar inequalities, there is no such inequality for single-particle systems. And it is for that reason that the logical possibility of hidden variables for single-particle systems is not excluded by Bell's theorem, as you suggested. So, there could be hidden variables under the hood, but if they are local and realistic (so the mainstream interpretation of Bell's assumptions goes), then those hidden variables may explain single-particle phenomena, but not the spooky results we get from entanglement.

A few things to say about that:

• No matter what, you can say that the universe does have phenomena which are not consistent with the classical idea of a local, realistic hidden variable theory (LRHVT). This on its own makes Bell's theorem interesting and shows that there's more to the universe than what people pictured before QM came about.
• Although you could hypothetically explain single-particle phenomena with some LRHVT, the reason that Bell asked himself "Could we have a LRHVT?" was to know if it would be possible in principle to find some kind of very down-to-earth explanation for entanglement, if we understood it better. And it turned out this was not possible. In contrast, the results of single-particle measurements don't beg as much for an explanation, because there is no apparent instantaneous effect over large distances, which Einstein originally claimed was in contradiction with relativity.
• Thanks for the answer! This clears it up. So there could be a local hidden variable theory for single particle phenomena (emphasis on could). This experiment just rules it out for entanglement. Commented Aug 19, 2022 at 23:41
• Yes, and it doesn't even rule it out for all entangled systems. Bell's theorem rules it out for the specific scenario he applied it to. Since then, people have applied a similar formula to many systems, but I don't think we're at the point where we can say that every entangled system has a Bell-like inequality. In fact, it might even be the case that we can show that there is no such inequality for certain systems - I would not be surprised, but I don't know. All it would take is a counter-example of a LRHVT for any particular entangled system which is able to correctly make predictions. Commented Aug 19, 2022 at 23:44
• One obvious counter-example is a system in which a single particle's position wave function is entangled with its spin wave-function, but this is kind of cheating because it's a single-particle system. Therefore "locality" can't be violated anyway... you're not comparing two distinct spacially separated points. Commented Aug 19, 2022 at 23:44
• Outside entanglement could be manyworlds : interpreting $A(a,\lambda)$ as "result of measurement apparatus A in direction of measure given by angle a 'in the world indexed by $\lambda$'" , maybe ? Commented Sep 25, 2022 at 13:14