Exercise 6.5

Take \(\sigma_z\) as an example. Let \(\sigma_z\) act on a product state: \[ \begin{flalign*} & \sigma_z ( \alpha_u \beta_u \ket{uu} + \alpha_u \beta_d \ket{ud} + \alpha_d \beta_u \ket{du} + \alpha_d \beta_d \ket{dd} ) = \\ & \alpha_u \beta_u \sigma_z \ket{uu} + \alpha_u \beta_d \sigma_z \ket{ud} + \alpha_d \beta_u \sigma_z \ket{du} + \alpha_d \beta_d \sigma_z \ket{dd} = \\ & \alpha_u \beta_u \ket{uu} + \alpha_u \beta_d \ket{ud} - \alpha_d \beta_u \ket{du} - \alpha_d \beta_d \ket{dd} \end{flalign*} \]

The result is again a product state.

This can be shown for any component of Alice’s or Bob’s spin operators: Look at Table 1 in the appendix. If any of the spin operators acts on the set of the product space basis vectors, it reproduces the same set of basis vectors, apart from the factors \(-1\), \(\mathrm{i}\) or \(-\mathrm{i}\). This doesn’t affect the general form of a product state.

For the second part of the exercise, let’s take again \(\sigma_z\) as an example and calculate the expectation value \(\braket{\sigma_z}\) for a single spin state \(\ket{\Psi_s} = \alpha_u \ket{u} + \alpha_d \ket{d}\): \[ \begin{flalign*} \braket{\sigma_z} &= \braket{\Psi_s | \sigma_z | \Psi_s} \\ &= \braket{ \bra{u} \alpha_u^* + \bra{d} \alpha_d^* | \sigma_z | \alpha_u \ket{u} + \alpha_d \ket{d} } \\ &= \braket{ \bra{u} \alpha_u^* + \bra{d} \alpha_d^* | \alpha_u \ket{u} - \alpha_d \ket{d} } \\ &= \alpha_u^* \alpha_u - \alpha_d^* \alpha_d \end{flalign*} \]

Now let’s calulate the expectation value for \(\sigma_z\) in a product state \(\ket{\Psi_p} = \alpha_u \beta_u \ket{uu} + \alpha_u \beta_d \ket{ud} + \alpha_d \beta_u \ket{du} + \alpha_d \beta_d \ket{dd}\): \[ \begin{flalign*} \braket{\sigma_z} &= \braket{\Psi_p | \sigma_z | \Psi_p} \\ &= \braket{\Psi_p | \alpha_u \beta_u \sigma_z \ket{uu} + \alpha_u \beta_d \sigma_z \ket{ud} + \alpha_d \beta_u \sigma_z \ket{du} + \alpha_d \beta_d \sigma_z \ket{dd} } \\ &= \braket{\Psi_p | \alpha_u \beta_u \ket{uu} + \alpha_u \beta_d \ket{ud} - \alpha_d \beta_u \ket{du} - \alpha_d \beta_d \ket{dd} } \\ &= \braket{ \alpha_u^* \beta_u^* \bra{uu} + \alpha_u^* \beta_d^* \bra{ud} + \alpha_d^* \beta_u^* \bra{du} + \alpha_d^* \beta_d^* \bra{dd} | \alpha_u \beta_u \ket{uu} + \alpha_u \beta_d \ket{ud} - \alpha_d \beta_u \ket{du} - \alpha_d \beta_d \ket{dd} } \\ &= \alpha_u^* \beta_u^* \alpha_u \beta_u + \alpha_u^* \beta_d^* \alpha_u \beta_d - \alpha_d^* \beta_u^* \alpha_d \beta_u - \alpha_d^* \beta_d^* \alpha_d \beta_d \\ &= \alpha_u^* \alpha_u (\beta_u^* \beta_u + \beta_d^* \beta_d) - \alpha_d^* \alpha_d (\beta_u^* \beta_u + \beta_d^* \beta_d) \\ &= \alpha_u^* \alpha_u - \alpha_d^* \alpha_d \end{flalign*} \]

The same pattern applies to other components of \(\sigma\) and \(\tau\).