I recently asked this question regarding type I and II fusions on polarization photons. To be honest, I am still not really clear how this operation generates cluster states, as the output at the end of the answer I was given was not an cluster state, but a 3 qubit GHZ state. While that may be unitarily equivalent, the literature is very clear that you end up with cluster states. As shown in the paper they start with pair of polarization bell states
$$\frac{1}{2} |H,H\rangle+|V,V\rangle \otimes |H,H\rangle+|V,V\rangle = \frac{1}{2} |H,H,H,H\rangle + |H,H,V,V\rangle + |V,V,H,H\rangle + |V,V,V,V\rangle$$
Then they send one mode from each pair into a PBS that reflects vertical light and then apply a detector after rotating the reflected polarization by 45 degrees, followed by a measurement. This rotation and detector are applied on only one of the two modes.
After the PBS, I believe you end up with
$$\frac{1}{2} |H,H,H,H\rangle+|H,HV,0,V\rangle+|V,0,VH,H\rangle + |V,V,V,V\rangle$$
From here, what I want to know is how this state is acted on by the rotation and detector, and how this results in the following cluster state
$$\frac{|H,+,H\rangle+|V,-,V\rangle}{\sqrt{2}} $$
If this optical component does in fact not give us the above state, then I would like some clarification on why these, and Type II fusions, are apparently used in paradigms like Fusion Based QC to create and manipulate cluster states, if they don't actually create or manipulate them. They can't just be creating GHZ states and doing local unitaries, as no paper I have read discusses this, and the effect measurements would have one them, due to the low persistency.
Edit: Going through the actual process of applying the PBS on subsystems 2 and 3, and then the 45 degrees rotation via HWP on subsustem 3, I get
$$\frac{1}{2} |H,H,\frac{H+V}{\sqrt{2}},H\rangle+\frac{1}{2} |H,HV,0,V\rangle+\frac{1}{2} |V,0,\frac{H-V}{\sqrt{2}}\frac{H+V}{\sqrt{2}},H\rangle +\\ \frac{1}{2} |V,V,\frac{H-V}{\sqrt{2}},V\rangle$$
$$=$$ $$\frac{1}{2\sqrt{2}} |H,H,H,H\rangle+\frac{1}{2\sqrt{2}} |H,H,V,H\rangle+\frac{1}{2}|H,HV,0,V\rangle+\frac{1}{4}|V,0,HH,H\rangle+\\\frac{1}{4}|V,0,HV,H\rangle-\frac{1}{4}|V,0,VH,H\rangle-\frac{1}{4}|V,0,VV,H\rangle + \frac{1}{2\sqrt{2}}|V,V,H,V\rangle-\\\frac{1}{2\sqrt{2}}|V,V,V,V\rangle$$
However, if you only detect a sinle photon, the state I believe you get is
$$\frac{1}{\sqrt{2}} |H,H,+,H\rangle+|V,V,-,V\rangle$$
However, from here, I am unsure how this process was meant to result in
$$\frac{|H,+,H\rangle+|V,-,V\rangle}{\sqrt{2}}$$
as they don't talk about a destructive measurement or discarding a subsystem.
