Exercise 7.4

Consider the state vector

\[ \ket{\Psi} = \alpha \ket{u} + \beta \ket{d} \]

The corresponding bra vector is: \[ \bra{\Psi} = \bra{u} \alpha^* + \bra{d} \beta^* \]

The density matrix \(\rho_{a, a'}\) is defined as

\[ \rho_{a, a'} = \braket{a | \Psi} \braket{\Psi | a'} \]

\[ \braket{u|\Psi} = \braket{u|\alpha|u}+\braket{u|\beta|d}=\alpha\braket{u|u} + \beta \braket{u|d} = \alpha \] \[ \braket{d|\Psi} = \braket{d|\alpha|u}+\braket{d|\beta|d}=\alpha\braket{d|u} + \beta \braket{d|d} = \beta \] \[ \braket{\Psi|u} = \braket{u|\alpha^*|u} + \braket{d|\beta^*|u}=\alpha^*\braket{u|u}+\beta^*\braket{d|u}=\alpha^* \] \[ \braket{\Psi|d} = \braket{u|\alpha^*|d} + \braket{d|\beta^*|d}=\alpha^*\braket{u|d}+\beta^*\braket{d|d}=\beta^* \]

\[ \rho_{u,u} = \braket{u | \Psi} \braket{\Psi | u} = \alpha \alpha^* \] \[ \rho_{u,d} = \braket{u | \Psi} \braket{\Psi | d} = \alpha \beta^* \] \[ \rho_{d,u} = \braket{d | \Psi} \braket{\Psi | u} = \beta \alpha^* \] \[ \rho_{d,d} = \braket{d | \Psi} \braket{\Psi | d} = \beta \beta^* \]

Hence \[ \rho_{a, a'} = \begin{pmatrix} \rho_{uu} & \rho_{ud}\\ \rho_{du} & \rho_{dd} \end{pmatrix} = \begin{pmatrix} \alpha \alpha^* & \alpha \beta^* \\ \beta \alpha^* & \beta \beta^* \end{pmatrix} \]

For \(\alpha = \beta = \frac{1}{\sqrt{2}}\) we get:

\[ \rho_{a,a'} = \begin{pmatrix} \frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} \]