Relation between Locality and Determinism

Question

In a previous question I asked, I was confused about how can you refuse determinism/realism in Bell's theorem without also refusing relativistic locality. I would like to understand where my following argument goes wrong.

Suppose that a physical theory is able to assign probabilities to events in space-time. The interesting thing that can happen is when two events $A$ and $B$ have related statistical distributions: $P(A) = f(P(B))$. If the function $f$ is invertible, then the events have a perfect statistical correlation.

Empirical science is built around the idea that we can explain correlations between events by means of "causes", by freely choosing an action $X$ to perform on a system $A$ we see if $P(A) = f(P(X))$ and deduce that $X$ has caused the outcomes. In a local theory, such causes (should they exist) are all in the past light-cone of $A$.

When Alice and Bob share an entangled state over space-like distances, and Alice freely chooses to orient the magnet at some angle, quantum mechanics predicts correlation that couldn't have possibly be caused by anything in the past light-cone of Alice and Bob.

Still, the outcome distribution that Bob sees is a function of how Alice freely chooses to orient the magnet, and this is normally considered a causal effect all over science. The fact that we can't observe it doesn't change this, so how exactly does giving up realism/determinism make everything local?

Answer 1

Before I get to Bell experiments, it’s important to clarify an important fact obscured by your set-up. Observed correlations, as described in your second paragraph, need not be the same as causal correlations, described in your third paragraph. This is a point that has taken people a long time to wrap their heads around, and has only become common wisdom in the past 20-30 years, as widely popularized by computer scientist Judea Pearl, and others. But it’s now an established fact that you can’t extract causal relationships from the “observed correlations” alone.

The key idea here is that there’s a difference between seeing and doing, a difference between observing a variable is set to X, and actually setting that variable to X. For instance, a barometer reading X is correlated with the weather, but if you physically manipulate the barometer and set that value to something else, you will break the correlation with the weather. Correlations that happen when you do something are causal relationships. In this example, evidently, we can’t cause the weather.

In the Bell experiment context, it’s certainly true that Alice’s setting (and Bob’s setting) are things that they do. Anything correlated with those settings are therefore effects, caused by Alice and Bob’s choice of settings. And the two outcomes taken as a pair are correlated with those settings. The pair of settings must therefore be a cause of those outcome pairs. And it’s not simply that they are only causing their own local outcome; that doesn’t work in this case. Something stranger is going on.

Still, this analysis gets tricky when you start fine-graining this causal relationship. If you talk about Bob’s outcome alone, and ask whether or not Alice’s setting caused it, it's hard to discuss without a bigger picture, without a full causal model. One very nice analysis of this, in the above causal-model framework, is due to Wood and Spekkens. They analyze a number of different causal models which might be responsible for the Bell experimental correlations (nonlocal, retrocausal, superdeterministic). Referring to those models might be a better way to frame your precise question.

Note that none of these approaches give up realism, and determinism doesn’t come into play either. Instead, the distinction is that in some causal models there are direct non-local links and in other models there aren’t. A retrocausal model, for instance, just has a causal link between Alice’s setting and the hidden variables in her past lightcone (and a similar link for Bob). Whether this model is “local” or not depends on your precise definition of the term, but realism and determinism aren’t the issues. The issue is the causal model.

Answer 2

When Alice and Bob share an entangled state over space-like distances, and Alice freely chooses to orient the magnet at some angle, quantum mechanics predicts correlation that couldn't have possibly be caused by anything in the past light-cone of Alice and Bob.

Still, the outcome distribution that Bob sees is a function of how Alice freely chooses to orient the magnet, and this is normally considered a causal effect all over science. The fact that we can't observe it doesn't change this, so how exactly does giving up realism/determinism make everything local?

The outcome distribution that Bob sees when he measures his system is not a function of how Alice orients her magnet. This can be seen by simply calculating the reduced density matrix of the system Bob is measuring. That density is not a function of anything Alice does.

If Bob compares his measurement results to those of Alice then he will see that they exhibit correlations that can't be explained by a local realistic theory (i.e. - a theory in which measurements have a single result described by a stochastic variable).

Quantum theory is not realistic in that sense. Systems, including measurement devices, are not described by stochastic variables, they can be described by Heisenberg picture observables that evolve locally and are represented by Hermitian operators. We get expectation values from those observables by combining them with the state using the Born rule.

When Alice and Bob do their measurements the observables of the measurement devices are affected locally by the system they measure. The measurement devices observables then depend on those of the system they measured: Alice's measurement device observables depend on those of Alice's measured system and likewise for Bob. There isn't a single outcome for the measurement: there is one outcome for each of the possible values. Each outcome is dynamically isolated from the others by decoherence:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

https://arxiv.org/abs/0707.2832

Then a further measurement is done that compares the measurement results from Alice and Bob's measurement devices. To do that measurement information must be sent to the comparison device and that information is described by the Heisenberg picture observables of the decoherent systems that are used to carry the information to the comparison device. Those observables carry information about both of the possible outcomes of the measurement. When the comparison happens each of Alice's outcomes is matched up with those for Bob and that is when the correlations arise:

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

https://arxiv.org/abs/1109.6223

https://arxiv.org/abs/2008.02328

The lack of realism helps because as a result of Alice and Bob's measurements having multiple outcomes the outcomes can be matched when the comparison is done rather than being matched at the time of the separate measurements, which would require some kind of unknown non-local process that nobody has ever described.