Bell Test Experiments

A Bell-test experiment compares a statistic computed from separated measurement records with the limit of a declared local model class. Its conclusion combines a mathematical inequality, a physical protocol and finite-data analysis. A measured violation is therefore stronger than observing entanglement alone, but it is not an assumption-free verdict about every possible explanation of reality.

From apparatus to conditional probabilities

A minimal record contains a trial identifier, settings $x,y$, outcomes $a,b$ and the timing information needed to justify the causal arrangement. The protocol specifies which events define trials, how losses are represented, how settings are generated and when data collection stops. These decisions determine whether the empirical frequencies estimate the probabilities appearing in the chosen inequality.

For sign outcomes one often reports conditional correlations

$$\widehat E_{xy}=\frac{N_{++}+N_{--}-N_{+-}-N_{-+}}{N_{xy}}.$$

This formula is meaningful only after the denominator’s trial population is defined. Dropping no-click trials without a justified rule can replace the intended population by a biased one. Bell Loopholes gives a concrete counterexample.

Three complementary landmark designs

The 2015 experiments below addressed the major locality and detection issues simultaneously in different physical systems. They are dated examples of experimental logic, not a ranking of current performance.

Experiment	Physical system	Design feature to study
Hensen and collaborators	Electron spins separated by 1.3 km	Event-ready heralding and efficient spin readout
Giustina and collaborators	Entangled photons	Efficient photon detection and rapid separated setting choices
Shalm and collaborators	Entangled photons	Loss-aware analysis, spacelike arrangement and explicit hypothesis testing

Read the primary methods and supplements when using any reported significance. A quoted Bell parameter alone does not identify the statistical model or show that its assumptions were implemented.

A simple statistical benchmark

For the uniform-input CHSH game, a Bell-local strategy has conditional win probability at most $3/4$ when fresh inputs are independent of the relevant prior information. In a fixed $n$-round experiment, a bounded-difference concentration bound gives the illustrative estimate

$$\Pr(\widehat w\ge3/4+\delta)\le e^{-2n\delta^2}.$$

This statement needs the stated conditional bound, fixed stopping and bounded score. It can allow memory through the conditional expectation formulation, but it must not be applied blindly to a ratio of separately estimated correlations or an adaptively stopped dataset.

For $n=1000$ and $\widehat w=0.80$, the bound is $e^{-5}\approx0.0067$. More efficient statistical methods often give different bounds; this calculation illustrates how trial count and excess score enter one valid benchmark, not the published analysis of any experiment in the table.

What a successful test establishes

A sufficiently significant violation rejects the specified Bell-local null under the experimental assumptions. It supports quantum predictions without uniquely selecting among all interpretations that reproduce those predictions. No-signaling is compatible with the observed Bell nonlocality, and claims about randomness or cryptographic security require further certification theorems.

For historical development see Bell-Inequality Experiments. The experimental roadmap compares Bell tests with other foundational tests.

Exercise: increasing statistical evidence

Keeping the same measured win fraction $0.80$ and the benchmark assumptions above, compare the exponential bound at $n=1000$ and $n=2000$.

Solution

The bounds are $e^{-5}$ and $e^{-10}$, about $0.0067$ and $0.000045$. Doubling the fixed trial count doubles the negative exponent. This does not license stopping at whichever sample first gives a desired bound; such a stopping strategy needs its own valid sequential analysis.

References

• Giustina, Marissa, et al., “Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons,” Physical Review Letters 115, 250401 (2015), doi:10.1103/PhysRevLett.115.250401.
• Hensen, B., et al., “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015), doi:10.1038/nature15759.
• Shalm, Lynden K., et al., “Strong Loophole-Free Test of Local Realism,” Physical Review Letters 115, 250402 (2015), doi:10.1103/PhysRevLett.115.250402.
• Zhang, Yanbao, Scott Glancy, and Emanuel Knill, “Asymptotically optimal data analysis for rejecting local realism,” Physical Review A 84, 062118 (2011), doi:10.1103/PhysRevA.84.062118.