Do Bell’s inequalities assume determinism?

Question

I was watching a video of Tim Maudlin where he talks about how the CHSH version of Bell’s inequalities do not assume determinism and only assume locality. He said that it is a common misconception that Bell assumed determinism and that in the CHSH version, he was explicit about not assuming that. He says there are statistical correlations in that version that are violated that apply to all local theories

So this implies, according to him, that physicists who say they can get out of Bell’s theorem by saying that the world is not deterministic don’t actually get out of it. Is this true?

Answer 1

Maudlin is correct.

In Bell’s original 1964 paper, he used a two-step argument. The first step (building off of the argument in the Einstein-Podolsky-Rosen (EPR) paper) got him to determinism, and the second step was the (new) argument that led from there to the consequence known as Bell’s Theorem. A cursory read of that paper might make people mistakenly think that he had assumed determinism, rather than deriving it from the EPR argument.

As Maudlin notes, later derivations of the Bell/CHSH theorem (by Bell and others) did not use this two-step structure. Starting from two basic assumptions (Locality and Causality, for short), and not assuming Determinism of any sort, Bell’s conclusions can be shown to go through in a single step.

Sadly, it was often claimed that one could get around Bell’s conclusion by simply allowing a theory with randomness, but fortunately those claims have subsided as of late. After all, if those claims were true, they certainly would not have given a Nobel Prize for the experimenters who showed the violation of Bell Inequalilities. If determinism was an assumption, all the Nobel Prize winners would have shown was that the world was inherently non-deterministic, rather than what they did show: that the world is either non-local or retrocausal.

Answer 2

The original Bell inequalities (1964) assume something that later was called "determinism" (or outcome determinism) which means that $$P(AB,xy\lambda)\in\{0,1\}\tag{1}$$ where $P(AB,xy\lambda)$ is the probability of that Alice measures $a$ and Bob measures $b$ given the configurations $x$ and $y$ of their detectors and hidden variable $\lambda$. It just means that if a hidden variable is given, only possible outcome is possible (something or nothing). How (1) ties to usual philosophical concept of determinism is a matter of interpretation. This condition was necessary along other two assumptions (statistical independence and locality).

However, when Bell wrote his Theory of local beables (1975), he came up with another assumption that he called local causality, which did not need the of condition (1). How local causality ties in to the philosophical meaning of determinism is also matter of interpretation.

Further reading: Speakable and Unspeakable in Quantum Mechanics by John S. Bell, Cambridge

See also: Bell theorem (Stanford Encyclopedia of Philosophy)

Comment: note that words like realism, determinism and locality are a matter of debate, looking for those in PhysicsSE would provide seemingly conflicting answers because not everyone agrees what is meant (for example locality in quantum field theory is most of the times, but not always, considered different from locality in Bell's inequality). However if you write down the equations, the requirements for the theorem are clear and is up to you to get a feeling of what is meant.

Answer 3

Bell's inequality predicts a value for the correlations of entangled pairs must be greater than a certain value, if there are predetermined hidden variables encoded in the particles at the time of emission. If the correlations dot exceed this value then then it predicts there can be be no predetermined hidden variables and that the outcome of measurements is determined at the time the measurement is made. This is condition is called a 'violation' of Bell's inequalities and it implies non local interactions (i.e .action at a distance exceeding the speed of light) between the particles.

Bell's paper introducing his inequalities in 1964 was a response (nearly 30 years later!) to a thought experiment called the EPR paradox, that was thought up (nearly 30 years earlier) by Einstein, Podolsky and Rosen in 1935, to demonstrate that Quantum Mechanics must be incomplete, because it appears to imply non local superluminal interactions, which of course Einstein would be very much against because his Relativity theory insists nothing, not even communication, can exceed the speed of light. Unfortunately for Einstein, later experiments confirmed that Bell's inequalities are violated and that non local interactions must be taking place between entangles pairs.

Does this invalidate Relativity? No. this is because the non local interactions occurring between distant particles at the quantum level, can not be used in any way by physical observes to send communication signals between themselves faster than light, nor can any physical interactions between macroscopic objects, or changes in physical fields, occur at greater than the speed of light.

Now to focus on the issue of determinism, the orientation of polarisation (or other observables) of the emitted entangled pairs is purely random. Bell's inequalities addresses the issue of whether these observables are fixed at the time of emission (pre-determined) or at the time of measurement. Bell predicted that they are determined at the time of measurement and subsequent experiments confirmed that prediction. In other words If the outcome of experiments find correlations that do not violate Bell's inequalities, then that concurs with the point of view that the correlations' are deterministic in the sense that they are determined at the time of emission, but experiments show that is not the case.

There are of course many interpretations of Quantum Mechanics, some such as the 'many worlds' interpretation that says every outcome occurs simultaneously and we just see one possible outcome and there is also the even more deterministic 'pilot wave' interpretation. Bell himself never really mentioned determinism in his paper, but focused on locality and hidden variables. If you consider hidden variables to be deterministic, then a non-violation of his inequalities would be a proof of determinism, but that has not been observed in reality.

Answer 4

Bell inequalities assume that measurement devices are not correlated with the means by which measurements are chosen and that physical systems are described by stochastic variables: numbers picked out of a set of possible values at random. A local non-deterministic theory respecting those constraints will respect the Bell inequalities, but experiments violate the Bell inequalities so such a theory doesn't describe reality.

Quantum theory without collapse describes physical systems in terms of observables, not stochastic variables:

https://arxiv.org/abs/quant-ph/9906007

https://arxiv.org/abs/1109.6223

The relevant equations of motion are local so the observables evolve locally and so quantum theory without collapse is a local theory that violates the Bell inequalities. While the evolution of observables is determined by their initial conditions this doesn't allow you to predict a single result you will get when you do an experiment, only the expectation value of the result. So whether you want to call it deterministic or not is up to you.