Bell's theorem simple limited proof

Consider the experiment illustrated in Figure 6.5. Two exactly correlated photons are created simultaneously. They move off in opposite directions for a distance that could be a billion light years in a thought experiment but must be considerably less in any practical experiment. Eventually each photon encounters an experimental setup consisting of a polarizing filter and a detector. Experimenters at each location can vary the angle of the polarizer.

Quantum mechanic implies that the two photons do not have a definite polarization until one of them is detected but once they are detected they will have identical polarizations. Thus quantum mechanics predicts that the probability of a joint detection is $\cos(\theta)$ where $\theta$ is the angle between the polarizers. In a local theory one can compute independent functions

and

that give the probability of detection of each particle as a a function of the angle of the local polarizer. Bell proved it was impossible to separate out the probabilities in this way. He derived an inequality by assuming the two detections were determined by some arbitrary local hidden variables. He showed quantum mechanics predicts the inequality is violated.

We will first give a simplified version of Bell's proof that assumes a particular hidden variables model based on an idea of d'Espagnat[12][11]. The more general proof is simple but requires more mathematics. We start by deriving one version of Bell's inequality. This derivation only involves logic. It has nothing to do with physics.

Consider three properties

and

that an object might have. The objects and properties could be anything. For example the objects could be words and the three properties could be whether a word contains the letter `a', `b' or `c'. Another example might be pictures containing the colors red, green and blue. Now consider three categories of objects: $(a \wedge \overline{b})$ , $(b \wedge \overline{c})$ and $(a \wedge \overline{c})$ . Assume we have a collections of objects that are candidates for each category. Denote the number of objects in a category by $N(a \wedge \overline{b})$ etc. The following must hold.

$\begin{displaymath} N(a \wedge \overline{b}) + N(b \wedge \overline{c}) \geq N(a \wedge \overline{c}) \end{displaymath}$

(6.1)

It is simple to prove Equation 6.1. Consider any object

that satisfies $(a \wedge \overline{c})$ . Now either

satisfies

or it does not. In the first case it also satisfies $(b \wedge \overline{c})$ and if it does not satisfy

it satisfies $(a \wedge \overline{b})$ . Every element counted in $N(a \wedge \overline{c})$ must also be counted in either $N(a \wedge \overline{b})$ or $N(b \wedge \overline{c})$ . Thus the above relationship holds.

This is one version of many results that are referred to as Bell inequalities. These are constraints that any hidden variables theory must satisfy. The proofs have nothing to do with quantum mechanics. The results depend only on logic and mathematics. However ``Bell's inequality'' always refers to a constraint which a hidden variables theory must satisfy and which quantum mechanics predicts will be violated.

Now we will see how quantum mechanics predicts equation 6.1 is violated. We cannot use polarized photons because we need some way of detecting that a particle does not have some property. Thus we use particle spin.

Some particles behave as if they were tiny spinning magnets. The spin is quantized it has a fixed amplitude in either a clockwise or counterclockwise direction. The assumption that the particles have definite spin is a local hidden variables model. It contradicts the quantum mechanical assumption that the state has no well defined value until it is observed. We will show how this assumption is inconsistent with the predictions of quantum mechanics.

When two particles are created together in an appropriate event they are said to be in a singlet state.Their spins when observed must be equal and opposite in direction. Depending on the orientation of the particles spin relative to a detector it will be deflected up or down. Its paired twin will have opposite spin and be deflected in the opposite direction. We say the particles are spin up or spin down.

We assume each particle has a definite spin before it is detected. We divide the particles into three categories.

One can determine if a particle is not detectable with spin up at a given angle by looking at the particle it is paired with. If a particle is detectable with spin up at some angle than the particle it is paired with cannot be because they have opposite spin. Thus we need to count the detections of paired particles

and

that fall into the following three categories in which all detections are with spin up.

These three categories fit the constraints of Equation 6.1. Quantum mechanics predicts that the probability that two particles will be detected at spin up with an angle $\phi$ between the detectors as follows.

$\begin{displaymath}\frac{1}{2}(\sin(\frac{\phi}{2}))^2\end{displaymath}$

$\begin{displaymath}\frac{1}{2}(\sin(\frac{45^\circ}{2}))^2 +\frac{1}{2}(\sin(\frac{45^\circ}{2}))^2 \geq \frac{1}{2}(\sin(\frac{90^\circ}{2}))^2 \end{displaymath}$

$\begin{displaymath}0.0732 + 0.0732 \geq 0.25 \end{displaymath}$

Clearly the local hidden variables model with the particles having a definite spin orientation before they are detected is wrong.

**Figure 6.5:** Experiment to test locality
$\begin{figure}\sf\begin{center} \raggedright\begin{picture}(400,80)(0,0) \par\pu... ...ions are consistent with relativity. \par\begin{center} \end{center}\end{figure}$