PDF version of this book

Bell's theorem simple limited proof

Consider the experiment illustrated in Figure 8.4. 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 $P(a)$ and $P(b)$ 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.

Following is a simplified version of Bell's proof that assumes a particular hidden variables model. It is based on an idea of d'Espagnat[17][16]. The more general proof is simple but requires more mathematics. These two derivations are strictly mathematical. They have nothing to do with physics. Physics enters the picture only in deriving predictions that violate the constraints derived in these proofs..

Consider three properties $a$, $b$ and $c$ 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.

$$N(a \wedge \overline{b}) + N(b \wedge \overline{c}) \geq N(a \wedge \overline{c})$$ (8.1)

The above and later equations are numbered to make them easy to refer to.

It is simple to prove Equation 8.1. Consider any object $x$ that satisfies $(a \wedge \overline{c})$. Now either $x$ satisfies $b$ or it does not. In the first case it also satisfies $(b \wedge \overline{c})$ and if it does not satisfy $b$ 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.

Quantum mechanics predicts equation 8.1 is violated. Polarized photons cannot be used in this simplified proof because it requires determining that a particle does not have some property. Particle spin does work.

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. 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. The particles are spin up or spin down.

Assume each particle has a definite spin before it is detected. Divide the particles into three categories.

1. Particles detectable with spin up at $0^\circ$ but not at $45^\circ$.
2. Particles detectable with spin up at $45^\circ$ but not at $90^\circ$.
3. Particles detectable with spin up at $0^\circ$ but not at $90^\circ$.

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. Count the detections of paired particles $x_i$ and $x_i'$ that fall into the following three categories in which all detections are with spin up.

1. $x_i$ detectable at $0^\circ$ and its pair $x_i'$ at $45^\circ$.
2. $x_i$ detectable at $45^\circ$ and its pair $x_i'$ at $90^\circ$.
3. $x_i$ detectable at $0^\circ$ and its pair $x_i'$ at $90^\circ$.

These three categories fit the constraints of Equation 8.1. Quantum mechanics predicts that the probability that two particles will be detected at spin up with an angle $\phi$ between the detectors as follows.

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

Substituting into Equation 8.1 gives the following.

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

Which evaluates as follows.

$$0.0732 + 0.0732 \geq 0.25$$

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

Figure 8.4: Experiment to test locality

PDF version of this book