Nielsen and Chuang: Solving equation of motion for amplitude damping

Question

I would like to know how to obtain a solution to the equation of motion given in Section 8.4.1 Master equations of Nielsen and Chuang, 10th edition.

The equation of motion that allows getting the quantum operation of amplitude damping is given by $$\frac{d \tilde{\rho}}{dt} = \gamma \left( 2 \sigma_-\tilde{\rho} \sigma_+ -\sigma_+ \sigma_- \tilde{\rho} - \tilde{\rho}\sigma_+ \sigma_- \right),$$ where $\sigma_- = |0\rangle \langle 1|$, $\sigma_+ = |1\rangle \langle 0|$ and $\tilde{\rho}(t) = e^{iHt} \rho(t) e^{-iHt}$.

In the book it is stated that using a Bloch vector representation, it is easy to get the solution: $$\tilde{\rho}(t) = \mathcal{E}(\tilde{\rho}(0)) = E_0\tilde{\rho}(0)E_0^{\dagger} + E_1\tilde{\rho}(0)E_1^{\dagger}$$ with $E_0 = \begin{pmatrix} 1 & 0\\ 0 & \sqrt{1 - \gamma'} \end{pmatrix}$ , $E_0 = \begin{pmatrix} 1 & \sqrt{\gamma}'\\ 0 & 0 \end{pmatrix}$ and $\gamma' = 1 - e^{-2 \gamma t}$.

What would be the steps to get the solution? The way it is written in the book, it should be easy.

Answer 1

Think about how the density matrix is represented with the Bloch vector: $$\tilde{\rho} = \frac{1}{2}(I + \vec{n} \cdot \vec{\sigma})$$ Then $$\frac{\mathrm{d}\tilde{\rho}}{\mathrm{d}t} = \dot{\vec{n}} \cdot \vec{\sigma} $$ is giving you the left hand side of the equation. We now want to find differential equations for the components of the Bloch vector $\vec{n}$. Therefore you'd need to plug the Bloch representation of the density operator into the rhs and subsequently take the inner product with the pauli-matrices (since $n_i = \mathrm{Tr}[\tilde{\rho}\sigma_i]$ for $i=x,y,z$). If this is not clear, look up the Hilbert-Schmidt inner product.

You'll find: $$\begin{aligned}\dot{n}_x &= -\gamma \, n_x \\ \dot{n}_y &= -\gamma \, n_y\\ \dot{n}_z &= 2\gamma \, (1- n_z)\end{aligned}$$. You can solve this and then validate that this time evolution is equivalent to applying the channel you mentioned.

Another option would be to solve this component wise for the 2x2 density matrix.

I hope this helps!