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Photon polarization states form a qubit $(\cos \theta ~ \sin \theta)^{T}$ - characterized by a parameter $\theta$. Obviously, such a state can be rotated by some transformation matrix $R_{\theta} = (\cos \theta, \sin \theta; -\sin \theta, \cos \theta)$. However, I do not understand what a polarization beamsplitter (PB) does to such a vector (probably splits it into two orthogonal vectors). What is the corresponding operator for PB, say $S_{\theta}$? How can one physically implement $R_{\theta}$ and $S_{\theta}$ (if it exists)?

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To discuss polarizing beam splitters you'll need to add another degree of freedom for the photon: its path. Initially, your photon has some polarization state and is incident in some path: $$|\psi\rangle=(\alpha|H,0\rangle+\beta|V,0\rangle)$$ where I labelled the path as 0. A polarizing beam splitter directs all photons of one polarization (say, horizontal) to one path and all photons of the other to an orthogonal path. For example $$PBS|\psi\rangle=(\alpha|H,2\rangle+\beta|V,3\rangle).$$ Now, if you inspect the photon in path 2, for example, you'll always find it to have polarization $|H\rangle$, but you'll only find it in that path with probability $|\alpha|^2$.

There are many question here, you can keep reading the quoted Wikipedia page for more answers. For example, you can make a PBS by "tilting a stack of glass plates at Brewster's angle to the beam." To make this into a polarizer that only transmits light of a particular polarization, you can ignore or absorb light exiting one of the two paths. To just change the polarization and do nothing to the path degree of freedom, you can use a waveplate; essentially a birefringent material that transmits different polarizations at different speeds such that they pick up a relative phase.

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