Exercise 7.9

The wave function \(\psi\) of a product state factorizes: \[ \psi(a,b) = \psi_A(a) \psi_B(b) \]

where \(\psi\) is the wave function of the composite system and \(\psi_A\), \(\psi_B\) are the wave functions of the sub systems. As an example, look at eq. (6.5) in the textbook:

\[ \ket{product \, state} = \alpha_u \beta_u \ket{uu} + \alpha_u \beta_d \ket{ud} + \alpha_d \beta_u \ket{du} + \alpha_d \beta_d \ket{dd} \]

The coefficients of the combined basis vectors are products of the coefficients of the sub systems’ basis vectors.

The expectation value of Alice’s observable \(\mathrm{A}\) is given by (see eq. 7.19 in the texbook) \[ \begin{flalign*} \braket{\mathrm{A}} &= \braket{\Psi | \mathrm{A} | \Psi} \\ & = \sum_{a', b, a}\psi^*(a', b) \, \mathrm{A}_{a', a} \, \psi(a,b) \\ &= \sum_{a', b, a} \psi^*_A(a') \psi^*_B(b) \, \mathrm{A}_{a', a} \, \psi_A(a) \psi_B(b) \\ &= \sum_{a', a} \psi^*_A(a') \, \mathrm{A}_{a', a} \, \psi_A(a) \end{flalign*} \] since the wave function \(\psi_B\) is normalized: \(\sum_b \psi^*_B(b) \psi_B(b) = 1\).

Likewise, the expectation value of Bob’s observable \(\mathrm{B}\) is given by \[ \begin{flalign*} \braket{\mathrm{B}} &= \braket{\Psi | \mathrm{B} | \Psi} \\ &= \sum_{b', a, b}\psi^*(a, b') \, \mathrm{B}_{b', b} \, \psi(a,b) \\ &= \sum_{b', a, b}\psi_A^*(a) \psi_B^*(b') \, \mathrm{B}_{b', b} \, \psi_A(a) \psi_B(b) \\ &= \sum_{b', b}\psi_B^*(b') \, \mathrm{B}_{b', b} \, \psi_B(b) \end{flalign*} \] since the wave function \(\psi_A\) is also normalized: \(\sum_a \psi^*_A(a) \psi_A(a) = 1\).

The expectation value of the product \(\mathrm{AB}\) is \[ \braket{\mathrm{AB}} = \braket{\Psi | \mathrm{AB} | \Psi} = \sum_{a',b',a,b}\psi^*(a', b') \, \mathrm{A_{a', a}}\, \mathrm{B_{b', b}} \, \psi(a,b) \]

Since the wave function of a product state factorizes, we can write this as: \[ \begin{flalign*} \braket{\mathrm{AB}} &= \sum_{a',a, b', b}\psi_A^*(a') \psi_B^*(b')\, \mathrm{A_{a', a}} \, \mathrm{B_{b', b}} \, \psi_A(a) \psi_B(b) \\ &= \sum_{a',a}\psi_A^*(a') \, \mathrm{A_{a', a}} \, \psi_A(a) \sum_{b',b}\psi_B^*(b') \, \mathrm{B_{b', b}} \, \psi_B(b) \\ &= \braket{\mathrm{A}} \braket{\mathrm{B}} \end{flalign*} \]

\[ \Rightarrow C(\mathrm{A}, \mathrm{B}) = \braket{\mathrm{AB}} - \braket{\mathrm{A}} \braket{\mathrm{B}} = 0 \]