Exercise 6.6

The expectation value in the singlet state is \[ \braket{\sigma_x \tau_y} = \braket{ sing | \sigma_x \tau_y | sing } \]

\[ \begin{flalign*} \sigma_x \tau_y \ket{sing} & = \sigma_y \tau_y \frac{1}{\sqrt{2}} ( \ket{ud} - \ket{du})&\\ & = \sigma_x \frac{1}{\sqrt{2}} (-\mathrm{i} \ket{uu} - \mathrm{i} \ket{dd} )&\\ & = -\mathrm{i} \frac{1}{\sqrt{2}} \sigma_x ( \ket{uu} + \ket{dd})&\\ & = -\mathrm{i} \frac{1}{\sqrt{2}} ( \ket{du} + \ket{ud}) \end{flalign*} \]

\[ \begin{flalign*} \braket{\sigma_x \tau_y} & = -\mathrm{i} \frac{1}{2} ( \bra{ud} - \bra{du})( \ket{du} + \ket{ud})\\ & = -\mathrm{i} \frac{1}{2} (1-1) = 0 \end{flalign*} \]

The statistical correlation is defined as \(\braket{\sigma_x \tau_y} - \braket{\sigma_x}\braket{\tau_y}\). As shown in the textbook (chapter 6.8), \(\braket{\sigma_x} = \braket{\tau_y} = 0\).

Thus the correlation is zero, which means the two measurements are not correlated.