How can a half-wave plate affect the position qubit and not the polarization qubit of a photon?

Question

I recently was watching these 2 videos on Coursera which show how to build a simple quantum computer that can implement the simplest case of the Deutsch-Jozsa algorithm (which uses only 2 qubits).
https://www.coursera.org/lecture/quantum-computing-algorithms/quantum-computer-prototype-diy-dCKRO
https://www.coursera.org/lecture/quantum-computing-algorithms/quantum-computer-prototype-solving-the-deutschs-problem-7EuD2
In it, the professor mentioned that the photon represented 2 qubits: one was the polarization qubit and one was the path/position qubit. Later, the professor stated that a half-wave plate affected the path/position qubit (not the polarization qubit). I found this really strange as I thought half waveplates only affected polarization.
How can a half-wave plate affect the position qubit and not the polarization qubit?

DaftWullie · Answer

The waveplates can be understood in terms of polarisation, but the change that they implement is independent of what polarisation the photon is in. It is simply "if the photon passes through this waveplate, x happens to it" where x might be "apply phase $e^{ipi}$".
Because there are two different paths, and you put a waveplate on a single path, the net effect is "if the photon traveled down that path, do x to it". Hence, it affects the position component.

How can a half-wave plate affect the position qubit and not the polarization qubit of a photon?

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