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

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?

• Are you interested in classical or quantum experiments? I can imagine taking a deck of cards, splitting them, and doing random measurements that could test the inequality. Oct 17, 2022 at 11:22
• An even more interesting question would be are there any classical experiments showing a VIOLATION of the Bell Inequalities? Oct 17, 2022 at 12:11
• So is the idea something like we take a single deck of cards, separate the cards into two piles and then separate the piles at such a distance they can't possibly influence each other, and then look at how the cards we draw from one pile are correlated with the other? Oct 18, 2022 at 3:43
• @peter-moore could you get a classical Bell Inequality violation somehow from polarized electromagnetic waves? Oct 18, 2022 at 3:48
• Yeah. what I'm getting at is, if you could devise a classical Bell-type experiment where local realism couldn't sanely be questioned, and yet consistently find inequality violations, wouldn't that force us to rethink all of this? Not to say the math is wrong, but our understanding of what local realism means would have to be called into question. I have to imagine no such experiments have found inequality violations or we'd have heard, but I'm going to ask a new question on this cause I'd really like to know. Oct 18, 2022 at 21:29

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.

• Do you have the formula for the locally real photon polarization that predicts a lower correlation? I want to know how its different from just using the QM answer of $P(a, b) = a \cdot b$. Is this dot product version what you mean when you say "entangled", and the "locally real" version just means making the assumption that the probability function is separable? Oct 16, 2022 at 20:38
• Bell’s original paper was published in a short-lived journal, and in the past was tough to get a copy of. But, living in the future has made finding things easier. The mathematics in the paper is quite accessible.
– rob
Oct 17, 2022 at 0:43
• so you don't have the formula for the locally real photon polarization. Answer is kinda incomplete without it, could you add it for completeness? Oct 18, 2022 at 3:54
• See expressions (10) and (11) in Bell’s paper. There are enough clarifying statements which precede them that they don’t really fit within this answer.
– rob
Oct 18, 2022 at 12:26
• I'm getting confused because on one hand you say to look at Bell's paper, but then you say that the equations in it don't relate to what you're saying here. I just want to understand exactly what Bell was assuming when he derived his inequality. Can you add any summary at all of what you're referring to beyond that its "quite accessible"? Oct 19, 2022 at 17:55

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.

• I can see how unentangled particles would "satisfy" Bell's inequality, but their correlation function is zero by definition, so all that says is 0 < 1 - a trivial statement that holds true regardless of the rotation of the detectors or the value of any hidden variables. But can Bell's inequality ever apply at all to an entangled system? Are there any particles where we expected the QM entanglement behavior, but they actually surprised us by satisfying the equality? Oct 16, 2022 at 20:29
• You could make a simulation of a 'quantum entangled system' that DOES have local realism applied, and that does not violate bell's inequality. What you do is, instead of measuring the spins of two entangled photons, you simply draw an angle on a circle of paper, and the opposite angle on another circle of paper. You treat that angle as the 'spin', and you make 360 pairs of such circles, or 180, or even just 9 pairs will be enough really. Then for each pair, you do the sorts of Spin measurements you would do in a bell test.
– TKoL
Oct 19, 2022 at 11:31
• And there you have it: a local hidden variable simulation of entangled photons.
– TKoL
Oct 19, 2022 at 11:32
• Just as an example of the 9 pairs of circles, pair 1 would be at 0° and 180° angle, pair 2 would be at 20° and 200°, pair 3 would be at 40° and 220°, then 60 and 240, 80 260, 100 280, 120 300, 140 320, and finally 160 340. You set up two 'detectors' to measure each one in the same way a bell test would, and you'll see the inequalities satisfied.
– TKoL
Oct 19, 2022 at 11:36
• @TKoL definitely starting to think that a classical experiment like this would show Bell's inequality being satisfied. How do you vary the "orientation" of the two detectors when using the strips of paper version? Oct 19, 2022 at 19:21