# 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 $$(1)$$?

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. Commented Jan 15, 2023 at 18:06
• But, looking at it, isn't it just some triangle inequalities? Commented Jan 15, 2023 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. Commented Jan 15, 2023 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
Commented Jan 15, 2023 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. Commented Jan 15, 2023 at 18:50

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 $$\pm (\epsilon+\delta)$$.
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 $$b-\eta \le a \le b+\eta$$.