Are the inequalities from Bell's Theorem ever actually satisfied?

Question

Following Griffiths's description from Introduction to Quantum Mechanics, Bell's original experiment was based on the EPR-Bohm experiment, where one considers the decay of a neutral pi meson into an electron and positron:

$$ \pi^0 -> e^{-} + e^{+} $$

Assuming the pion is at rest, the electron and positron will fly off in opposite directions. There are two detectors that will measure the spin of the electron and the positron. Since the pion has spin 0, the electron and positron must have opposite spins, and the spins measured by the two detectors are thus correlated. This is an example of the famous Einstein, Podolsky, and Rosen or EPR Paradox.

Bell suggested a version of this experiment where the two detectors are allowed to be rotated independently. The first detector measures the component of the electron spin in the direction of a unit vector $\boldsymbol{a}$, and the second measures the spin of the positron along the direction $\boldsymbol{b}$. Bell proposed to measure the average value of the product of the spins. Call this average $P(\boldsymbol{a}, \boldsymbol{b})$. If the detectors are parallel then $\boldsymbol{a} = \boldsymbol{b}$ and we recover the configuration of the EPR-Bohm experiment above. In the general case, Bell derives the following inequality for a local hidden variable theory in the variable(s) $\lambda$:

$$ |P(\boldsymbol{a}, \boldsymbol{b}) - P(\boldsymbol{a}, \boldsymbol{c})| \leq \int{p(\lambda)[1 - A(\boldsymbol{b}, \lambda) A(\boldsymbol{c}, \lambda)] d\lambda} $$

Or, more simply:

$$ |P(\boldsymbol{a}, \boldsymbol{b}) - P(\boldsymbol{a}, \boldsymbol{c})| \leq 1 + P(\boldsymbol{b}, \boldsymbol{c}) $$

I'm looking for experimental evidence showing that this Bell inequality is ever satisfied. The only experiments I know of have shown this type of inequality being violated, hence the deductions that no such local hidden variable theory can exist. But what evidence is there showing that this type of inequality can be satisfied?

I know that the experiment trivially satisfies the inequality when $\boldsymbol{a} \cdot \boldsymbol{b}$ is equal to either $0$, $+1$, or $-1$. Are there any experiments where $0 < |\boldsymbol{a} \cdot \boldsymbol{b}| < 1$ and a Bell inequality is still satisfied?

Answer 1

Consider the more practical example of entangled photon emission, as from parametric down-conversion. From the details of the photon production process, you know that the photons will have orthogonal plane polarizations in any basis. If both of your detectors are in the horizontal-vertical orientation, you’ll see evidence of perfect orthogonality. If both detectors are in the diagonal-antidiagonal basis, you’ll also see evidence of perfect orthogonality. Bell’s theorem is interesting for the case where one detector is in the horizontal-vertical basis and the other detector is in the diagonal-antidiagonal basis. In that case, a “locally real” photon polarization predicts a smaller correlation than a quantum-mechanical entanglement of the two photon polarizations.

You can destroy the entanglement by analyzing the spin of the photons before they reach your detector. Suppose each photon passes through a polarizing filter before it reaches the detector. The disentangling filters are always perpendicular, though their orientation relative to the detection analyzers is random.

Bell’s theorem says that, when the detectors analyze the polarization is different bases, the disentangled locally-real photon pairs will have less correlation than the entangled pairs. The inequality is between the locally-real correlation versus the entangled correlation.

Note that Bell’s paper has two sections. The first, more commonly cited, finds that quantum correlations are larger than the specific model of “perhaps the spins are real but not known.” The second section shows that any hidden-local-variable will produce smaller correlations than quantum entanglement, but it doesn’t preclude a hypothetical hidden-local-variable model which is more strongly correlated than the specific example from the first section. Many papers on Bell’s inequality treat correlations larger than “the locally-real prediction” as “proof of entanglement,” which is not quite right.

I don’t know whether the experiment with disentangled pairs has been done intentionally and published. It might have been part of Aspect’s early experimental work.

It might also appear in the literature of quantum encryption. In quantum encryption, two observers conduct random measurements on entangled pairs, then share their detector orientations afterwards on a possibly-insecure channel. For measurements where their detectors used the same basis, the anticorrelation can be used to transmit information. For measurements where one observer analyzed horizontal-vertical and the other analyzed diagonal-antidiagonal, a Bell-inequality violation guarantees that no eavesdropper has measured their entangled photon pairs and surreptitiously replaced them. A leader in that literature has been Zeilinger, who (with Aspect and Clauser) has just won a Nobel.

Answer 2

The inequality is satisfied by any experiment that satisfies the assumptions of Bell's proof, which would include, for instance, spin measurements of unentangled particles.

If the particles are halves of two different Bell pairs, then the result of a measurement along any axis is $\pm 1$ with equal probability, so the averages are all $0$ and the inequality is satisfied regardless of the axes.

You probably won't find an experiment that looks for non-violation of the inequality in cases where quantum mechanics predicts non-violation, since there's no point in doing an experiment for which all theories predict the same result. I don't know of any non-classical, non-quantum theory that would predict violation of the inequality where quantum mechanics doesn't.