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

enter image description here

(from here)

enter image description here

enter image description 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?
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You get entanglement if you use single photons (or other types of nonclassical light) instead of coherent light.

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.

As to whether it's legitimate to consider single photons as "entangled", I'll remind you to the discussion in this other post. The gist being that yes, you arguably can, and a standard reference here is (S. J. van Enk 2005).

As to how would a CNOT gate be implemented in this setup?: the PBS is a CNOT in this position-polarization encoding. Specifically, it flips the position state conditionally to the polarization.

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