Does Bell's theorem imply a causal connection between the measurement outcomes?

Question

It is a standard result that quantum mechanics does not allow for superluminal communication, which would seem to directly imply that the answer to this question is no. After all, Bell's result tells us about correlations, and we know that correlation does not necessarily imply causation.

However, if we restrict our attention to solely the measurement outcomes, Bell's theorem also tells us that we cannot think of the observations of Alice and Bob as correlated via a third variable. In other words, it rules out the following causal structure:

More precisely, the result (stated in the CHSH formulation) is that, if there is an underlying probability distribution $p(a,b,x,y,\lambda)$ which is such that (*) $$p(a,b|x,y,\lambda)=p(a|x,\lambda)p(b|y,\lambda),\tag1$$

then some functional relations between the values of $x, y$ and the conditional marginal distribution $p(a,b|x,y)$ are not possible. In other words, (1) imposes restrictions on the following function: $$f(a,b,x,y)\equiv \sum_\lambda p(a,b|x,y,\lambda)p(\lambda)=p(a,b|x,y).\tag2$$ These restrictions are made evident when taking expectation values over all possible outcomes, that is by studying the function $g$ defined as $$g(x,y)\equiv\sum_{a,b}ab \,\,f(a,b,x,y).$$

In particular, it is not possible (and this is the content of CHSH result) to have something like $$g(x,y)=\boldsymbol x\cdot\boldsymbol y.$$

Now let us consider the quantum case. The full probability distribution has to be written in the more general form $p(a,b,x,y)$, and whenever a form of Bell's inequalities is violated by this distribution, then we know that we cannot factorize it with the help of an additional variable as in (2).

However, we can still assume that the measurement setups are chosen independently, so that we can still write $$p(x,y)=p(x)p(y),$$ althought we cannot write $p(a,b,x)=p(a,x)p(b)$ or $p(a,b,y)=p(b,y)p(a)$.

My question is then precisely about these last statements: does the fact that $p(a,b,x)$ cannot be factorized as $p(a,x)p(b)$ imply that there is a causal relation between the two measurement outcomes?

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(*)

Eq. (1) can be equivalently stated as a statement about the structure of the full joint probability distribution describing $p(a,b,x,y,\lambda)$ the whole behavior: $$p(a,b,x,y,\lambda)=\frac{p(x,y,\lambda)}{p(x,\lambda)p(y,\lambda)}p(a,x,\lambda)p(b,y,\lambda),$$ where every time a variable is not included as an argument we are talking about the marginal distribution with respect to that variable, so that for example: $$p(a,x,\lambda)\equiv\sum_{b,y}p(a,b,x,y,\lambda).$$

Answer 1

It kind of does, but in a useless way.

The question is essentially equivalent to the following simplified version of it: suppose a probability distribution $p(a,b)$ cannot be factorized as $p(a)p(b)$. Does this imply that $A$ "causes" $B$?

The answer is: not really. The problem is that, from a purely probabilistic point of view, there are no "causal relations", only correlations. Whenever there are correlations between variables, their marginals can be written so that one variable "looks caused" by the other, but this statement does not have much value.

Indeed, we can always write $p(a)=\sum_b p(a|b)p(b)$, which makes $A$ being "caused" by $B$, because $p(a|b)$ is defined as $p(a|b)\equiv p(a,b)/p(b)$, and the marginal $p(a)$ by $p(a)\equiv\sum_b p(a,b)$. While true, this is not a very useful observation.

It is meaningful to talk about causation in a context in which one can actually exploit such correlation. For example, if one can choose to have the variable $B$ assume the value $b$, then it is meaningful to talk about the conditional probabilities $p(a|b)$. Notably, this is exactly the kind of thing that we cannot do in quantum mechanics: the measurement results are probabilistic, and therefore cannot be controlled.

Answer 2

You asked:

Does Bell's theorem imply a causal connection between the measurement outcomes?

Bells Theorem is essentially about measuring outcomes on an experiment on the EPR paradox. This is the paradox where Einstein argued that QM was not complete because entangled systems showed that superluminal signalling was possible - contra special relativity. Bell himself wrote:

If [a hidden variable theory] is local it will not agree with [[standard]] quantum mechanics, and if it agrees with [[standard]] quantum mechanics it will not be local. This is what the theorem says. Speakable and Unspeakable in Quantum Mechanics, John Bell

and he concluded:

In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant.

A hidden variable theory means simply a deterministic theory. What this means that given a deterministic theory of QM where cause propagates locally, and this means physically and hence at or less than the speed of light, then it won't match all the predictions of standard QM. Alternatively, if it turns out this deterministic theory does match the predictions, then cause will propagate faster than light speed.

Now since the predictions of standard QM have been verified to a high degree of accuracy then Bells theorem shows that QM is

• not deterministic, or

• deterministic with cause propagating at faster than light speeds

Generally the second option is ruled out on the basis of relativity and this leaves the first option - that QM is not deterministic. What this means about entangled systems being non-local is left open. However, its shown through the no communication theorem of QI that no actual effective information can be signalled at faster than light speeds. So QM remains local in this second, weaker sense of locality.

note: [[my insertions]]

Answer 3

What is really important to understand before formulating any answers is the exact meaning of the words that enters into the context of Bell's theorem and in particular locality, realism and causality.

Locality and causality are linked together in what is defined as "relativistic local causality".

Then it is unfortunate that realism - please let me quote wikipedia here - "is now somewhat different from what it was in discussions in the 1930s. It is more precisely called counterfactual definiteness".

I'm adding the previous sentence about the meaning of realism because the final answer to the OP question - which I want to rephrase as: does causality hold or not? - depends on which of the two - realism or locality - one chooses to abandon after Bell's theorem.

For example R. D. Gill suggests that we should "relinquish not locality, but realism": in that case we would somehow arrive to the conclusion of the OP's answer, i.e. that "the question of causality is ill-posed in this context".

On the other hand, I would prefer the opposite choice, arriving to the opposite conclusion, that causality is meaningful and QM (or De Broglie–Bohm or any other theory which satisfies Bell's inequality) has to be non-local theory (important note, non-local in the sense of Bell's theorem, but of course we all agree about the signal locality or no-signaling principle or no-communication theorem).

Finally, a minor note should be added about the fact that the experimental verification of the inequality is subject to the closure of all the possible loopholes (free will, fair sampling and so on).

Answer 4

does the fact that $p(a,b,x)$ cannot be factorized as $p(a,x)p(b)$ imply that there is a causal relation between the two measurement outcomes?

From a mathematical point of view that means that QM violates "outcome independence" . Together with the fact that $a$ and $b$ can be spacelike separated, this implies that QM violates local causality. This is not to be confused neither with the non-deterministic character of the theory nor with the possibility to exploit this feature to send FTL signals. Signal locality is indeed expressed by a different mathematical constraint: $p(a|x, y) = p(a|x)$ or equivalently $p(b|x, y) = p(b|y)$.

Nonetheless - whatever interpretation one chooses to explain the so-called collapse of the wave function - this kind of non-locality remains. For example Bohmian mechanics satisfies outcome Independence and fails to satisfy setting Independence. Back to QM, Bell's inequalities follow from the algebra of Hamilton's quaternions (more technically Bell quantum bound is obeyed for the case when four commutators vanish and Bell's inequality can be given meaning in the formalism of C*-algebras ). Interestingly there is also a geometric interpretation of the hidden variables as the global phase curled up under the Hopf-fibration between the unit spinor and the Bloch vector - which is again (without dwelling into the proof check of any alternative theory) a non-local topological mapping.

Answer 5

My question is then precisely about these last statements: does the fact that p(a,b,x) cannot be factorized as p(a,x)p(b) imply that there is a causal relation between the two measurement outcomes?

No it doesn't imply any causal relation between the measurement outcomes. Bell's theorem sez that any hidden variable theory must be non-local to match the predictions of quantum mechanics. Those hidden variables are descriptors of a system that have a single measurable value. Quantum mechanics is not a hidden variable theory, so the Bell inequalities don't imply that quantum mechanics is non-local.

Rather, each system exists in multiple versions and those versions are matched up in the appropriate way when the measurement results from the entangled systems interact. For example, if particle 1 is entangled with particle 2, and particle 1 is measured at a point p1 spacelike separated from the measurement of particle 2 at point p2, then all of the possible outcomes happen for each measurement happen. When these outcomes are compared at some third point p3 after being coveyed there by local means, the matching takes place, but before that comparison they are not matched. This matching is done using quantum information that each system holds that can't be extracted by measurements on that system alone and so doesn't suffer decoherence.

For a detailed quantum mechanical account of how the Bell correlations happen see David Deutsch, Patrick Hayden, 'Information Flow in Entangled Quantum Systems', Proc. R. Soc. Lond. A 456(1999):1759-1774. available at http://arxiv.org/abs/quant-ph/9906007. And also David Deutsch, 'Vindication of quantum locality', Proc. R. Soc. A 468(2012), 531-544. available at http://arxiv.org/abs/1109.6223.

This account is given using what is commonly called the "many worlds interpretation of quantum mechanics", which is just quantum mechanics taken literally as a description of how the world works. For reasons that are not well explained, this is controversial. There other "interpretations", some of which are alternative physical theories like the spontaneous collapse theory or the pilot wave theory. These are alternatives to quantum mechanics, not interpretations and they are in general non-local because they say that EPR type experiments have single outcomes. Other "interpretations" such as the Copenhagen or statistical interpretations don't make clear statements about what exists in reality and so no account can be given of what is happening in reality under those theories.