# Bell's Original Paper - Local hidden variable theories correlations smaller than entanglement

I'm having trouble following Bell's derivation of equation 22 in his original paper. Particularly, how to go from

$$| \overline{P} (\vec{a}, \vec{b}) - \overline{P}(\vec{a}, \vec{c}) | \leq 1 + \overline{P}(\vec{b}, \vec{c}) + \varepsilon + \delta$$

to

$$| \vec{a} \cdot \vec{c} - \vec{a} \cdot \vec{b} | - 2 (\varepsilon + \delta) \leq 1 - \vec{b} \cdot \vec{c} + 2(\varepsilon + \delta)$$

using

$$$$| \overline{P}(\vec{a}, \vec{b}) + \vec{a} \cdot \vec{b} | \leq \varepsilon + \delta. \qquad(1)$$$$

My attempt was

$$$$\begin{split} &\overline{P}(\vec{b}, \vec{c}) + \vec{b} \cdot \vec{c} \leq | \overline{P}(\vec{b}, \vec{c}) + \vec{b} \cdot \vec{c} | \leq \varepsilon + \delta\\ \implies & \overline{P}(\vec{b}, \vec{c}) \leq (\varepsilon + \delta) - \vec{b} \cdot \vec{c}\\ \implies & 1 + (\varepsilon + \delta) + \overline{P}(\vec{b}, \vec{c}) \leq 1 - \vec{b} \cdot \vec{c} + 2(\varepsilon + \delta)\\ \implies& | \overline{P} (\vec{a}, \vec{b}) - \overline{P}(\vec{a}, \vec{c})| \leq 1 - \vec{b} \cdot \vec{c} + 2(\varepsilon + \delta) \end{split}$$$$

How do I deal with the left hand side, i.e., the absolute value of difference of probabilities using $$(1)$$?

Edited to add more info

We are dealing with a pair of spin one-half particles in a singlet state moving freely in opposite directions.

$$P (\vec{a}, \vec{b}) \equiv \int d \lambda \rho(\lambda)A(\vec{a}, \lambda) B(\vec{b}, \lambda)$$

where $$\lambda$$ is a continuous parameter, $$\rho(\lambda)$$ is its probability distributions and $$A(\vec{a}, \lambda) = \pm 1$$, $$B(\vec{a}, \lambda) \pm 1$$ are the possibles results of measuring spin components $$\vec{\sigma}_{1}, \vec{\sigma}_{2}$$ along $$\vec{a}, \vec{b}$$ respectively.

The probabilities $$\overline{P}$$s are nonnegative, the vectors $$\vec{a}, \vec{b}, \vec{c}$$ are unit vectors and I'm assuning $$\varepsilon, \delta > 0$$.

• You will need to add details. What is P, how is it defined? You can't expect people to go and read the paper for you. Jan 15 at 18:06
• But, looking at it, isn't it just some triangle inequalities? Jan 15 at 18:08
• I wasn't expecting people to go and read the paper, I was hoping that someone who read the paper could help. Will add more information, but I guess it's not the right place for this kind of specific questions about a specific paper. Jan 15 at 18:17
• I remember having and answering this question when I read this paper, but I no longer have my notes from 2005. I'll try and write an answer later.
– rob
Jan 15 at 18:33
• @rob As far as I can tell you just insert (1) into the first line and get the second (in fact, with one factor of 2 only being 1). As far as I can tell this is homework-type. Jan 15 at 18:50

## 1 Answer

What Equation (1) says is that if you replace $$P(\vec x,\vec y)$$ by $$-\vec x\cdot \vec y$$, you incur an error of at most $$\pm (\epsilon+\delta)$$.

Now do this both on the rhs and the lhs of your first equation:

1. On the lhs of the "$$\le$$", in the worst case the error will make the value larger, which means you need to put a minus sign to still satisfy the inequality. You do two replacements, thus $$-2(\epsilon+\delta)$$.

2. On the rhs, in the worst case the error will make the value smaller. Thus, you get an extra $$+(\epsilon+\delta)$$.

If you want to formalize this, this can be done by the triangle inequality, plus the following variant thereof: If $$|a-b|\le \eta$$, then $$b-\eta \le a \le b+\eta$$.