What These Experiments Do and Do Not Prove

Foundations experiments constrain specific explanations of measurement statistics. Their conclusions become clearer when we identify the measured quantity, the model being tested, and the assumptions connecting that model to a bound. This page compares those conclusions and explains why strong correlations alone are insufficient.

Which Correlations Test Which Assumptions?

Experiment	What is measured?	What makes the conclusion possible?
Bell test	Correlations across several settings at separated stations	Local factorization and independent settings imply a bound; detection, timing, and statistics must match the tested scenario
Contextuality test	Statistics for overlapping compatible measurement contexts	The same measurement must have a context-independent representation; incompatibility and detection errors must be controlled
Delayed choice	Path-basis or interference-basis outcomes chosen after entry	The late choice implements a different final observable; predictions use the complete preparation and measurement arrangement
Quantum eraser	Joint path-system and marker outcomes	Conditional fringes depend on marker correlations; averaging over the marker removes the selected pattern

These are different tests. A double-slit fringe is not a Bell violation, and the absence of a fringe does not establish that the preceding state was an incoherent mixture. The detector may simply measure a basis insensitive to relative phase.

Perfect Anticorrelation Can Be Classical

Imagine a source drawing a fair random number $r=+1$ or $-1$. Alice receives $A=r$ and Bob receives $B=-r$, regardless of their settings. Every pair is perfectly anticorrelated, and each side alone sees equally many plus and minus results.

Nevertheless, for the four CHSH setting pairs,

$$E(a,b)=E(a,b')=E(a',b)=E(a',b')=-1,$$

so

$$S=E(a,b)+E(a,b')+E(a',b)-E(a',b')=-2.$$

The model saturates the local bound; it does not violate it. The singlet becomes a Bell test only when different analyzer directions produce a pattern of correlations that no local model with independent settings can reproduce. Suitable directions give $\lvert S\rvert=2\sqrt2$.

A general local model may have stochastic responses rather than predetermined binary values:

$$P(A,B\mid x,y)=\int d\lambda\,\rho(\lambda) P(A\mid x,\lambda)P(B\mid y,\lambda).$$

Here $\rho(\lambda)$ must be independent of the selected $x,y$. Violations exclude this structure under the experimental assumptions. They do not exclude hidden-variable theories that reject local factorization or setting independence.

Joint Correlations and Local Communication

No-signaling means that Alice’s marginal distribution does not depend on Bob’s setting:

$$\sum_B P(A,B\mid x,y)=P(A\mid x).$$

This condition is weaker than Bell-local factorization. Quantum measurements on separated subsystems satisfy it even when the joint distribution violates a Bell inequality. Alice cannot choose which random outcome Bob obtains. To study their correlations, the observers compare their records through an ordinary communication channel.

The same distinction resolves the marker-measurement quantum eraser. A marker outcome can identify a fringe or an antifringe subset, but the local observer who lacks that outcome sees their sum. Discarding knowledge of the marker does not restore coherence. Coherently undoing the marking interaction or applying a communicated phase correction is a different physical operation.

What Remains an Interpretive Question?

The apparatus can mark paths with polarization, recoil, or an internal state without anyone reading the marker. Entanglement and conditioning explain the resulting statistics without a consciousness-dependent dynamical rule. These experiments therefore do not demonstrate that a mind causes collapse.

They also do not uniquely select an interpretation of quantum mechanics. Interpretations with the same operational predictions agree on these tests. Theories that change the dynamics, such as objective-collapse models, can make further experimentally distinguishable predictions and must be tested against those predictions. See Measurement Problem as Historical Problem for the remaining issue of definite outcomes.

A mathematical theorem establishes a bound exactly from assumptions. A finite experiment supplies a statistical comparison with that bound. For example, a measured CHSH value slightly above $2$ is not by itself a significance statement: trial selection and statistical uncertainty still matter. Loophole-Free Bell Tests explains how experiments connect the ideal scenario to actual detections.

Exercises

1. In the shared-random-bit model above, Bob reverses his recorded sign only for setting $b'$. Recalculate the four correlations and $S$.

Solution

The correlations are $-1,+1,-1,+1$ in the order appearing in $S$. Thus $S=-1+1-1-1=-2$. Changing a local output convention does not turn this local model into a violation.

2. An experiment reports $S=2.10$ with an estimated standard error $0.15$. Is the point estimate enough to establish a statistically strong violation?

Solution

No. The excess is only $0.10/0.15\approx0.67$ standard errors in that approximate analysis. A justified significance claim also requires the statistical model and trial rules; the point estimate alone is insufficient.

3. Why does Bell’s theorem not disprove determinism by itself?

Solution

The tested model combines local factorization with setting independence. Deterministic and stochastic local models satisfy the bound. A deterministic theory with nonlocal dynamics, such as Bohmian mechanics, rejects an assumption of that model and is not excluded merely for being deterministic.

References

• A. Aspect, J. Dalibard, and G. Roger, “Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers,” Physical Review Letters 49, 1804-1807, 1982, DOI: 10.1103/PhysRevLett.49.1804.
• J. S. Bell, “On the Einstein Podolsky Rosen Paradox,” Physics 1, 195-200, 1964, DOI: 10.1103/PhysicsPhysiqueFizika.1.195.
• J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge University Press, 2004.
• N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419-478, 2014, DOI: 10.1103/RevModPhys.86.419.
• B. Hensen et al., “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682-686, 2015, DOI: 10.1038/nature15759.
• S. Kochen and E. P. Specker, “The problem of hidden variables in quantum mechanics,” Journal of Mathematics and Mechanics 17, 59-87, 1967.
• X.-S. Ma, J. Kofler, and A. Zeilinger, “Delayed-choice gedanken experiments and their realizations,” Reviews of Modern Physics 88, 015005, 2016, DOI: 10.1103/RevModPhys.88.015005.
• N. D. Mermin, “Hidden variables and the two theorems of John Bell,” Reviews of Modern Physics 65, 803-815, 1993, DOI: 10.1103/RevModPhys.65.803.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• M. Schlosshauer, “Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics,” Reviews of Modern Physics 76, 1267-1305, 2005, DOI: 10.1103/RevModPhys.76.1267.