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By definition from Wikipedia, the beam splitter (BS) operation $U_{BS} = \begin{pmatrix} t & r \\ r & t \end{pmatrix} = \begin{pmatrix} \cos\theta & -i\sin\theta \\ -i\sin\theta & \cos\theta \end{pmatrix}$, and the rotation matrix $R=\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$. Now a mirror is just a BS with $r=1$. Therefore it seems that $U_{Mirror} = \begin{pmatrix} \cos\theta & 1 \\1 & \cos\theta \end{pmatrix}$. However, it is mentioned on the same Wikipedia page that $U_{Mirror} = R$!

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Now a mirror is just a BS with $r=1$. Therefore it seems that $U_{Mirror} = \begin{pmatrix} \cos\theta & 1 \\1 & \cos\theta \end{pmatrix}$.

This is incomplete. If $|r|=|\sin(\theta)|=1$, then $\theta=\pi/2$ and $\cos(\theta)=0$, so $$U_\mathrm{mirror} = \begin{pmatrix} 0 & 1 \\1 & 0 \end{pmatrix}.$$

The Wikipedia page's assertion that the unitary corresponding to a mirror is a rotation matrix $R(\theta)$ with variable $\theta$ is incorrect.

That said, the details of what unitary needs to be used to describe a mirror will depend on exactly what labelling convention you use, so they cannot be given up front without that specification.

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