# Photon polarization transformations

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)?

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 $$|\alpha|^2$$.