# Operation Elements for Amplitude Damping Channel

To find operation elements for the Amplitude Damping channel, Nielsen and Chuang (in Section 8.3.5 of my copy) use the action of a beamsplitter on an initial state $$\alpha |0\rangle + \beta |1\rangle$$. The output of the beamsplitter is $$\alpha|0_E0\rangle + \beta \cos\theta |0_E1\rangle + \beta \sin\theta|1_E0\rangle$$ where $$E$$ denotes environment. They say after tracing out the environment we get $$E_0\rho E_0^\dagger + E_1\rho E_1^\dagger$$ where

$$E_0 =\begin{pmatrix}1 & 0 \\ 0 & \sqrt{1-\gamma}\end{pmatrix}$$,

$$E_1 =\begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0\end{pmatrix}$$

with $$\gamma = \sin^2\theta$$.

Here is what I do:

\begin{align*}tr_E(\rho^\prime) &= tr_E(\alpha^2|0_E\rangle \langle0_E|\otimes|0\rangle \langle0| + \beta^2 \cos^2\theta |0_E\rangle \langle0_E|\otimes|1\rangle \langle1| + \beta^2 \sin^2\theta|1_E\rangle \langle1_E|\otimes|0\rangle \langle0|) \\&= \alpha^2|0\rangle \langle0| + \beta^2 \cos^2\theta|1\rangle \langle1| + \beta^2 \sin^2\theta|0\rangle \langle0|\end{align*}

Another approach I tried:

$$\langle0_E|B|0_E\rangle = \alpha|0\rangle + \beta \cos\theta|1\rangle$$

which gives

$$\rho^\prime = \alpha^2|0\rangle \langle0| + \beta^2 \cos^2\theta|1\rangle \langle1|$$

and $$\begin{pmatrix}1 & 0 \\0 & \sqrt{1-\gamma}\end{pmatrix}$$ for $$E_0$$;

and

$$\langle1_E|B|0_E\rangle = \beta \sin\theta|0\rangle$$ with $$\rho^\prime = \beta^2 \sin^2\theta|0\rangle \langle0|$$

which gives

$$E_1 \rho E_1^\dagger= \begin{pmatrix}\beta^2\gamma & 0 \\0 & 0\end{pmatrix} = \begin{pmatrix}0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix} \rho \begin{pmatrix}0 & 0 \\ \sqrt{\gamma} & 0 \end{pmatrix}$$

Am I correct? If not, where am I wrong?

Here is a picture of the page for your reference:

• The density matrix from which you trace over are missing some terms. The will be terms like $\alpha^*\beta\cos(\theta)$ and so on... Some of these terms will go to zero when you take trace but some are left. – Hemant Jul 20 '19 at 1:57
• @Hemant Just so you're aware, you can Latex in comments! I've edited yours to add the Latex - hope you don't mind! (although it could be expanded into an answer...) – Mithrandir24601 Jul 20 '19 at 11:52
• @Mithrandir24601 Thanks for the edit. – Bashir Jul 20 '19 at 15:23
• Mith, Thanks for the edit. @Bashir There are some cross terms that DO NOT go to zero as well. – Hemant Jul 20 '19 at 17:32
• Yes, you are right. There are non zero cross terms. – Bashir Jul 21 '19 at 1:17