What Does Bell's Theorem Rule Out?

• Highest correlation a local theory allows
• Correlation measured in the experiment
• Bell limit (red dashed line)
• Point above the limit
• Correlation axis

What you will learn

• State the local condition that Bell’s theorem rejects and why it concerns causality.
• Describe a Bell test and how it blocks the loopholes.
• Tell apart the options that remain open after the theorem.

Before you start

It helps to read Can Entanglement Send Messages Faster Than Light? first. The terms hidden variable, Bell inequality and local are explained as they appear.

What does Bell’s theorem say?

Bell’s theorem says that no theory satisfying Bell locality, plus some supplementary assumptions, can reproduce all the probabilistic predictions of quantum mechanics. According to the Stanford Encyclopedia of Philosophy, that condition requires any correlation between distant events to be explicable in local terms, by states of affairs at the common source of the particles.

The theorem has roots in Bell’s investigation of the hidden-variables program. That program supplements the quantum state with further “elements of reality”, or hidden variables. You can picture each particle leaving the Stanford source with an instruction card that the measurement simply reads out.

From the local condition, one derives the Bell inequalities. Correlations from any theory meeting the conditions of the proof must obey them. Yet under certain circumstances, correlations calculated from quantum mechanics violate them.

Why does this touch causality? Bell said the scientific attitude is that correlations cry out for explanation. The Stanford entry adds that the explanation must be causal: either a causal connection between the two variables or a common cause. Reichenbach named the common-cause assumption the principle of the common cause. Bell’s theorem tests exactly this kind of explanation.

• Common source
• Instruction card each particle carries
• Two detectors
• Direct link ruled out (crossed dashed line)

What do Bell tests find?

Bell tests find results at odds with local theories. In 1964, according to a NIST news release, Bell showed there are limits to measured correlations that can be ascribed to local, pre-existing conditions. An experiment is a way to see whether nature exceeds those limits.

NIST researchers and several other institutions created pairs of identical photons and sent them to two locations to be measured. The two detectors are 184 m apart. While the photons are in flight, a random number generator picks one of two polarization settings for each analyzer.

Random choice and an early finish are deliberate. Information traveling at the speed of light would need 617 ns to travel between the detectors. The detectors finished more than 40 ns before that moment, so one could not have “told” the other its result in time.

Earlier experiments were vulnerable to at least one of two loopholes, according to Stanford: the communication, or locality, loophole and the detection loophole. In 2015, experiments demonstrated violation of Bell inequalities with those loopholes blocked. The NIST team simultaneously closed all three major loopholes that had plagued previous Bell tests.

NIST’s Krister Shalm put the point to remember this way: you cannot prove quantum mechanics, but local realism, or hidden local action, is incompatible with the experiment.

The 2022 Nobel Prize in Physics went to Alain Aspect, John Clauser and Anton Zeilinger. The prize cited experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science.

• Photon-pair source
• Two detectors
• Random setting picker (dice)
• Short bar: time taken by the measurement
• Long bar: time for light to travel between the detectors

After Bell’s theorem, which options remain?

After Bell’s theorem, at least one assumption of the proof has to go. The Stanford entry lays out the structure of the choice: the factorizability condition is the conjunction of outcome independence (OI) and parameter independence (PI). Rejecting factorizability means rejecting PI or OI.

The first road is to reject PI. That brings rejecting causal locality, which means accepting causal relations between events outside each other’s light cones. The de Broglie–Bohm theory takes this road: it violates PI but does not permit superluminal signalling.

The second road is to keep causal locality. Then OI has to go, and with it the Common Cause Principle.

Another option is to reconsider measurement independence. Random generators and spacetime arrangements help control the choice loophole, but do not prove absolute independence from every hidden factor. Superdeterminism is one rejection of that assumption discussed in the Stanford source.

NIST summarizes the choice this way: correlations beyond the Bell limits would require either sending signals faster than light, which scientists consider impossible, or another mechanism such as quantum entanglement.

• Result: violation of the Bell inequalities
• Give up causal locality
• Keep locality, give up the common-cause principle
• Hypotheses outside the test (dashed outline)

Real-world example

Imagine two people running lotteries in two cities whose results always match. Two explanations come to mind. One is that the two phone each other with the numbers. The other is that a single person printed a list of numbers in advance and gave a copy to each.

Bell’s theorem is a test for the second explanation. If the printed list were every cause there is, the correlation would have to stay under a limit. The experiment measures a correlation above the limit, with no phone line fast enough to join the two places.

This example illustrates the reasoning and is not a detailed account of the real experiment. The real experiment measures photon polarization.

Common misunderstandings

Bell’s theorem does not prove that entanglement sends messages faster than light. It concerns the limits of local explanations, and sending messages is a separate question.

A Bell test does not prove quantum mechanics correct. It rules out a class of local theories.

Hidden variables are not ruled out altogether. The de Broglie–Bohm theory is a hidden-variable theory that violates PI and still does not permit superluminal signalling.

Frequently asked questions

What does Bell’s theorem rule out?

It rules out theories that meet Bell locality plus some supplementary assumptions, because they cannot reproduce all the probabilistic predictions of quantum mechanics.

What is a loophole in a Bell test?

A weakness that leaves the result inconclusive. The two best known are the communication loophole and the detection loophole, and in 2015 experiments blocked them.

Are hidden variables still viable?

Local hidden variables are not, given the experimental results. The de Broglie–Bohm theory is still discussed because it rejects PI instead of keeping locality.

Does a Bell test prove quantum mechanics?

No. The researchers say plainly that quantum mechanics cannot be proven, only local realism ruled out.

Summary

• Bell’s theorem rejects theories that explain correlations only by states at the common source, in a local way.
• A loophole-free test with detectors 184 m apart gave results at odds with that class of theories.
• After the theorem, something has to go: causal locality, the common-cause principle, or an assumption such as free choice of settings.