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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.

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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!

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  • $\begingroup$ Thanks for the answer. It is definitely helpful! However, I don't think the $\frac{d}{dt} n_z = 2 \gamma (1-n_z)$ is correct cause its solution doesn't match the book's solution. Also, while you provided expressions for differential equations (with one equation likely being incorrect), you didn't answer my question about how/why $\tilde{\rho}(t) = \mathcal{E}(\tilde{\rho}(0)) = E_0\tilde{\rho}(0)E_0^{\dagger} + E_1\tilde{\rho}(0)E_1^{\dagger}$ is a solution. If you could improve your answer, that would be of great help. In either case, thank you again! $\endgroup$
    – MonteNero
    Dec 3, 2022 at 2:13
  • $\begingroup$ (1) Ok, no worries. I do not see the problem for the z component though. This ODE can be solved via separation of variables. As intermediate step this will give $n_z(t) = c \, e^{-2 \gamma t} +1$, where $c$ is the integration constant. However, $n_z(t=0) = n_z(0)$ and we can conclude that $c = n_z(0) - 1$. I believe that this is the solution in the book. (But you should :) just differentiate the book's solution and check). Regarding the question "why", I'm not sure that I can answer that (hopefully someone else though). IMO the way you formulate it, is a little bit unprecise. $\endgroup$ Dec 3, 2022 at 7:23
  • $\begingroup$ (2) What is stated in the book, is that the solution obtained from the Master equation is equivalent to applying the quantum channel $\mathcal{E}$. The master equation is an evolution equation for the density operator including a dissipative part (spontaneous emission). The point is that this evolution of the density operator, $\tilde{\rho}(t)$, can equivalently be obtained by applying the amplitude damping channel to the initial state $\tilde{\rho}(0)$. If you read the end of the section, it'll be explained that the quantum operation formalism is more general though. I hope this helps! $\endgroup$ Dec 3, 2022 at 7:24

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