# 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? 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. 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
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
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
Aug 23, 2022 at 16:59

• 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 ? Sep 25, 2022 at 13:14