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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. We repeat the treatment with additional commentary from Appendix 2 of Bell's article "Bertlmann's socks and the nature of reality"[6]. The commentary is intended to make the argument understandable even for those allergic to mathematical notation.

For the general case we need two adjustments $a$ and $b$ and two observations $A$ and $B$ as shown in Figure 6.5. We assume there is some unknown set of hidden variables that we represent by a single parameter $\lambda$. The assumption of locality is that we can factor the joint probability into two independent local probabilities. The joint probability of getting 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.

\begin{displaymath} P(A,B\vert a.b.\lambda) = P_1(A\vert a,\lambda)P_2(B\vert b,\lambda) \end{displaymath} (6.2)

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. We can remove $\lambda$ from the left hand side by integrating or averaging over all possible values of $\lambda$. To do this we need the probability density function $f(\lambda)$ that gives the likelihood that $\lambda$ will have a particular value.

\begin{displaymath}P(A,B\vert a.b) = \int P_1(A\vert a,\lambda)P_2(B\vert b,\lambda) f(\lambda)d\lambda\end{displaymath}

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 we can replace outcomes $(A,\,B)$ with 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.

$\displaystyle E(a,b)\hspace{.07in}=$ $\textstyle P(yes,yes\vert a,b) - P(yes,no\vert a,b)$   (6.3)
  $\textstyle - P(no,yes\vert a,b) + P(no,no\vert a,b)$    

So instead of computing the probability of possible outcomes $P(A,B\vert a,b)$ we will be computing 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 we get detections in only one of the two sensors rather than in both or neither.

We can write $E$ as a function of local variables using Equation 6.2. 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.

\begin{displaymath} E(a,b) = \int\{P_1(yes\vert a,\lambda) P_2(yes\vert b,\lambda) \end{displaymath} (6.4)

\begin{displaymath}- P_1(yes\vert a,\lambda) P_2(no\vert b,\lambda) \nonumber\end{displaymath}

\begin{displaymath}-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\end{displaymath}

We can factor the part of 6.4 involving $P_1$ and $P_2$ as follows.

\begin{displaymath} E(a,b) = \int{\{P_1(yes\vert a,\lambda) -P_1(no\vert a,\lambda)\}} \end{displaymath} (6.5)

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

We can rewrite 6.5 as follows.

\begin{displaymath} E(a,b) = \int \overline{A}(a,\lambda)\overline{B}(b,\lambda) f(\lambda) d\lambda \end{displaymath} (6.6)

By making the following substitutions.

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

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

Now we use 6.6 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 we will 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 $a -b -c +d$.

\begin{displaymath} E(a,b) \pm E(a,b') = \int \overline{A}(a,\lambda) \{\overlin... ...(b,\lambda) \pm \overline{B}(b',\lambda)\}f(\lambda) d\lambda \end{displaymath} (6.7)

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

$\displaystyle 0 \leq P_1 \leq 1$     (6.8)
$\displaystyle * 0 \leq P_2 \leq 1$      

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 6.7 by converting the equality to an inequality. It is convenient to do generate two versions of the result.

\begin{displaymath} \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 \end{displaymath} (6.9)

\begin{displaymath}\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\end{displaymath}

Using 6.8 we have the following.

\begin{displaymath} \vert\overline{B}(b,\lambda) \pm \overline{B}(b',\lambda)\ve... ...verline{B}(b,\lambda) \mp \overline{B}(b',\lambda)\vert \leq 2 \end{displaymath} (6.10)

Since $f(\lambda)$ is a probability density we have the following.

\begin{displaymath} \int f(\lambda) d\lambda = 1 \end{displaymath} (6.11)

Now using 6.11 and 6.10 we have the following from 6.9.

\begin{displaymath} \vert E(a,b) \pm E(a,b')\vert + \vert E(a',b) \mp E(a',b')\vert \leq 2 \end{displaymath} (6.12)

This is called the CHSH inequality from the initials of the authors who first derived it[9] as an inequality that could be tested experimentally.

For the experiment involving magnetic spin in Section 6.4 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 we have from 6.12 the following.

\begin{displaymath}\vert\cos(a-b) \mp \cos(a-b')\vert + \vert\cos(a'-b) \pm \cos(a'-b')\vert \leq 2\end{displaymath}

Assume $a = 0$, $b = 90^\circ$ and $a' = 45^\circ$. Figure 6.6 gives a plot of 6.12 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 6.6: 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.

Completed second draft of this book

PDF version of this book
next up previous contents
Next: Experimental tests of Bell's Up: Relativity plus quantum mechanics Previous: Bell's theorem simple limited   Contents

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