Exercise 6.1

The statistical correlation is defined as \[ \braket{\sigma_A \, \sigma_B} - \braket{\sigma_A} \braket{\sigma_B} \]

The expectation values are defined as usual: \[ \begin{flalign*} \braket{\sigma_A} &= \sum_{i=1}^n a_i \, P_A(a_i) \\ \braket{\sigma_B} &= \sum_{j=1}^m b_j \, P_B(b_j) \end{flalign*} \]

where the sum goes over all possible values of \(a\) and \(b\). The expectation value for the combined measurement is defined as \[ \braket{\sigma_A \, \sigma_B} = \sum_{i=1}^n \sum_{j=1}^m a_i \, b_j \, P(a_i, b_j) \]

Hence the correlation is \[ \begin{flalign*} \braket{\sigma_A \, \sigma_B} - \braket{\sigma_A} \braket{\sigma_B} &= \sum_{i=1}^n \sum_{j=1}^m a_i \, b_j \, P(a_i, b_j) - \sum_{i=1}^n a_i \, P_A(a_i) \cdot \sum_{j=1}^m b_j \, P_B(b_j) \\ &= \sum_{i=1}^n \sum_{j=1}^m a_i \, b_j \, P(a_i, b_j) - \sum_{i=1}^n \sum_{j=1}^m a_i \, b_j \, P_A(a_i) P_B(b_j)\\ &= \sum_{i=1}^n \sum_{j=1}^m a_i \, b_j \, (P(a_i, b_j) - P_A(a_i) P_B(b_j)) \\ &= 0 \end{flalign*} \]

if \(P(a_i, b_j) = P_A(a_i)\, P_B(b_j)\).