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I am trying to understand how the correlation function in John Bell's paper on EPR is derived for a spin singlet state $|{\Psi}\rangle$. This is defined to be

$$ \langle{\Psi}|\left(\bf{\sigma}\cdot\bf{a}\right)\left(\bf{\sigma}\cdot\bf{b}\right)|{\Psi}\rangle=-\bf a \cdot\bf b. $$ I tried to compute it explicitly by using the Pauli matrices, but was unable to derive the scalar product of the two direction vectors.

Edit

One attempt to prove this is by using $$ \langle{\Psi}|\left(\bf{\sigma}\cdot\bf{a}\right)\left(\bf{\sigma}\cdot\bf{b}\right)|{\Psi}\rangle=\langle{\Psi}|(\bf {a} \cdot\bf{ b})\bf I +i(\bf a \times\bf b)\cdot\sigma|{\Psi}\rangle $$ However, I am not able to derive the minus sing or get rid of the cross product term either. See the comment section

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  • $\begingroup$ You used the formula for the product of two Pauli vectors? $\endgroup$ – Cosmas Zachos Feb 24 at 15:26
  • $\begingroup$ Yes, but I was not able to figure out the $i(\bf{a}\times\bf{b})\cdot\bf{\sigma}$ part $\endgroup$ – Alexander Cska Feb 24 at 15:41
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You can solve this by applying the definition of the Pauli vector, i.e.,

$${\bf{\sigma}\cdot\bf{a} }=\sigma_xa_x+\sigma_ya_y+\sigma_za_z$$ where $a_x,a_y,a_z$ are the components of $\bf a$ and $\sigma_x,\sigma_y,\sigma_z$ are the Pauli matrices. I will assume that we apply ${\bf{\sigma}\cdot\bf{a} }$ on the first subsystem, and ${\bf{\sigma}\cdot\bf{b} }$ on the second subsystem.

If we use the standard notation for Bell states, i.e., $$|\psi^\pm\rangle=\frac{|01\rangle \pm|10\rangle}{\sqrt{2}}\quad;\quad|\phi^\pm\rangle=\frac{|00\rangle \pm|11\rangle}{\sqrt{2}}$$

You can easily verify, that $${\bf{\sigma}\cdot\bf{a} }|\psi^+\rangle=-a_x|\phi^-\rangle+ia_y|\phi^+\rangle+a_z|\psi^+\rangle$$ $${\bf{\sigma}\cdot\bf{b} }|\psi^+\rangle=b_x|\phi^-\rangle-ib_y|\phi^+\rangle+-b_z|\psi^+\rangle$$ Therefore, \begin{align} \langle\psi^+|({\bf{\sigma}\cdot\bf{a} })({\bf{\sigma}\cdot\bf{b} })|\psi^+\rangle&=\left[-a_x\langle\phi^-|-ia_y\langle\phi^+|+a_z\langle\psi^+|\right]\left[|b_x|\phi^-\rangle-ib_y|\phi^+\rangle+-b_z|\psi^+\rangle\right]\\&=-a_xb_x-a_yb_y-a_zb_z=-{\bf a\cdot b} \end{align} where the second equality follows from the orthogonality of Bell states, and the last one follows from the definition of scalar product.

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