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Bell's theorem general proof

This is the only section with substantial mathematics. It is not essential. One can understand the idea of Bell's result from the previous section. However it is worth making the effort to follow if you are at all inclined to do so.

Bell's proof that no hidden variables model will work involves some simple integral equations. The following repeats the treatment with additional commentary from Appendix 2 of Bell's article "Bertlmann's socks and the nature of reality"[7]. The commentary is intended to make the argument understandable even for those allergic to mathematical notation.

For the general case, two adjustments $a$ and $b$ and two observations $A$ and $B$ as are needed. These are shown in Figure 8.4. Assume there is some unknown set of hidden variables represented by a single parameter $\lambda$. The assumption of locality is that the joint probability of simultaneous detections factors into two independent local probabilities. The joint probability of detections $A$ and $B$ with experimental settings $a$ and $b$ is $P(A,B\vert a.b.\lambda)$. The locality assumptions is given by the following.

$$P(A,B\vert a.b.\lambda) = P_1(A\vert a,\lambda)P_2(B\vert b,\lambda)$$ (8.2)

This implies that the probability of a detection at $A$ is only dependent on the local setting of $a$ and a local unknown variable $\lambda$. The same is true at the other detector. $\lambda$ can be removed from the left hand side by integrating or averaging over all possible values of $\lambda$. To do this requires the probability density function $f(\lambda)$ that gives the likelihood that $\lambda$ will have a particular value.

$$P(A,B\vert a.b) = \int P_1(A\vert a,\lambda)P_2(B\vert b,\lambda) f(\lambda)d\lambda$$

This says that the joint probability of observing $A$ and $B$ at settings $a$ and $b$ is the average value of a product of three terms. The first two are the likelihood of making observations $A$ and $B$ locally for a given value of $\lambda$. The third $f(\lambda)$ is the likelihood that $\lambda$ will have this value.

Practical experiments detect or fail to detect a particle. Thus outcomes $(A, B)$ cam replace the four possible detection combinations of $(yes, yes)$, $(yes, no)$, $(no, yes)$ and $(no, no)$ representing a detection or no detection at the two distant sensors. The following sum of these possible outcomes is particularly useful in developing a version of Bell's inequality that can be tested experimentally.

$$E(a,b)\hspace{.07in}= P(yes,yes\vert a,b) - P(yes,no\vert a,b)$$ (8.3)

$$- P(no,yes\vert a,b) + P(no,no\vert a,b)$$

So instead of computing the probability of possible outcomes $P(A,B\vert a,b)$ compute the probability that $E(a,b)$ will have a given value. $E$ is not a probability and will be negative if it is more likely that there are detections in only one of the two sensors rather than in both or neither.

Write $E$ as a function of local variables using Equation 8.2. The idea is to replace the nonlocal dependency in $(a,b)$ with dependency only on local values (only $a$ or only $b$) and the hidden local variable $\lambda$. This results in the following.

$$E(a,b) = \int\{P_1(yes\vert a,\lambda) P_2(yes\vert b,\lambda)$$ (8.4)

$$- P_1(yes\vert a,\lambda) P_2(no\vert b,\lambda) \nonumber$$

$$-P_1(no\vert a,\lambda)P_2(yes\vert b,\lambda) + P_1(no\vert a,\lambda) P_2(no\vert b,\lambda)\}f(\lambda) d\lambda$$

Factor the part of 8.4 involving $P_1$ and $P_2$ as follows.

$$E(a,b) = \int{\{P_1(yes\vert a,\lambda) -P_1(no\vert a,\lambda)\}}$$ (8.5)

$$\times\{P_2(yes\vert b,\lambda) -P_2(no\vert b,\lambda)\}f(\lambda) d\lambda$$

Rewrite 8.5 as follows.

$$E(a,b) = \int \overline{A}(a,\lambda)\overline{B}(b,\lambda) f(\lambda) d\lambda$$ (8.6)

By making the following substitutions.

$$\overline{A}(a,\lambda) = \{P_1(yes\vert a,\lambda) -P_1(no\vert a,\lambda)\}$$

$$\overline{B}(b,\lambda) = \{P_2(yes\vert b,\lambda) -P_2(no\vert b,\lambda)\}$$

Use 8.6 to construct a formula for $E(a,b) \pm E(a,b')$. The symbol `$\pm$' is used as a shorthand for two equations. The first uses $+$ and the second $-$. You have to make the substitution every place $\pm$ occurs. Shortly Use $\mp$ which substitutes the signs in the reverse order. For example $a\pm b \pm c \mp d$ represents the two equations $a + b + c - d$ and $a -b -c +d$.

$$E(a,b) \pm E(a,b') = \int \overline{A}(a,\lambda) \{\overlin... ...(b,\lambda) \pm \overline{B}(b',\lambda)\}f(\lambda) d\lambda$$ (8.7)

Since $P_1$ and $P_2$ are probabilities the following holds.

$$0 \leq P_1 \leq 1$$ (8.8)

$$* 0 \leq P_2 \leq 1$$

Thus $\vert\overline{A}(a,\lambda)\vert \leq 1$ and $\vert\overline{B}(b,\lambda)\vert \leq 1$. For these are both differences between two numbers between 0 and 1 and thus their absolute value must be less than or equal one. Using this can remove $\overline{A}(a,\lambda)$ from 8.7 by converting the equality to an inequality. It is convenient to do generate two versions of the result.

$$\vert E(a,b) \pm E(a,b')\vert \leq \int \vert\overline{B}(b,\lambda) \pm \overline{B}(b',\lambda)\vert f(\lambda) d\lambda$$ (8.9)

$$\vert E(a,b) \mp E(a,b')\vert \leq \int \vert\overline{B}(b,\lambda) \mp \overline{B}(b',\lambda)\vert f(\lambda) d\lambda$$

Using 8.8 gives the following.

$$\vert\overline{B}(b,\lambda) \pm \overline{B}(b',\lambda)\ve... ...verline{B}(b,\lambda) \mp \overline{B}(b',\lambda)\vert \leq 2$$ (8.10)

Because $f(\lambda)$ is a probability density, the following must hold.

$$\int f(\lambda) d\lambda = 1$$ (8.11)

Applying 8.11 and 8.10 to 8.9 yields the following.

$$\vert E(a,b) \pm E(a,b')\vert + \vert E(a',b) \mp E(a',b')\vert \leq 2$$ (8.12)

This is called the CHSH inequality from the initials of the authors who first derived it[13] as an inequality that could be tested experimentally.

For the experiment involving magnetic spin in Section 8.4 provides a good example of how this inequality is violated. Quantum mechanics predicts that that $E(a,b) = -\cos(a-b)$ where $a$ and $b$ are the angles of orientation of the two polarizers. Thus from 8.12 we have the following.

$$\vert\cos(a-b) \mp \cos(a-b')\vert + \vert\cos(a'-b) \pm \cos(a'-b')\vert \leq 2$$

Assume $a = 0$, $b = 90^\circ$ and $a' = 45^\circ$. Figure 8.5 gives a plot of 8.12 as a function of $b'$. It also plots the classical limit of 2. The peak occurs at $135^\circ$ where the value for CHSH predicted by quantum mechanics is $2.828$.

Figure 8.5: Correlation predictions

The horizontal straight line at 2 represents the maximum correlation from local hidden variables theories. The curve is the prediction of quantum mechanics.

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