How to entangle 2 qubits using a polarizing beam splitter?

A polarizing beam splitter splits a continuous beam of photons in 2 directions:

(from here)

(from here)

I am curious and have a few related questions about using simple optics in quantum computing:

1. Is splitting a light beam by polarization a true quantum (probabilistic) effect?

2. How would one tensor two qubits using this setup to form the tensored/entangled state

$$(a|0\rangle + b|1\rangle)(c|0\rangle + d|1\rangle)= ac|00\rangle + ad|01\rangle + bc|10\rangle + bd|11\rangle$$

1. How would a CNOT gate be implemented in this setup?

Linear optical operations on coherent states never result in entanglement. Roughly speaking, a coherent state $$|\alpha\rangle$$ passing through a beamsplitter gives a state of the form $$|t\alpha\rangle|r\alpha\rangle$$ for some $$t,r$$, $$|t|^2+|r|^2=1$$. Similar results hold for arbitrary linear operations (linear here means operations that preserve the photon number). With a PBS, the same thing applies, except output modes also have a different polarization state, and so you can write them as something like $$|t\alpha_H\rangle|r\alpha_V\rangle$$. Point being, these are not entangled states.
Note that things are different with single photons, as the evolution then reads $$(a_H+a_V)\to a_{1,H}+a_{2,V}$$ where $$1,2$$ label different position states. Or in ket notation, you can write the latter state as $$|1,H\rangle+|2,V\rangle$$, which clearly looks entangled.