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?

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