Why does a BBO crystal create entangled photons?

If we pump a beta barium borate (BBO) crystal, we get one circle of vertically polarized photons | V ⟩ |V⟩ and an intersecting circle of horizontally polarized ones | H ⟩ |H⟩: http://quantum.ustc.edu.cn/old/img/image002.gif At the intersection points of the circles, we get entanglement | ψ ⟩ = 1 √ 2 ( | H ⟩ | V ⟩ + | V ⟩ | H ⟩ ) . |ψ⟩=12(|H⟩|V⟩+|V⟩|H⟩). Why do we get an entangled state and not just a mixed state ρ = 1 2 ( | H ⟩ | V ⟩ ⟨ H | ⟨ V | + | V ⟩ | H ⟩ ⟨ V | ⟨ H | ) ? ρ=12(|H⟩|V⟩⟨H|⟨V|+|V⟩|H⟩⟨V|⟨H|)? I know we can confirm the entanglement in many experiments, but I guess somebody first had to come up with the idea that this actually creates entanglement. As a related question, in this case there is no relative phase shift, but I’ve also read papers where the BBO created | ψ ⟩ = 1 √ 2 ( | H ⟩ | V ⟩ − | V ⟩ | H ⟩ ) , |ψ⟩=12(|H⟩|V⟩−|V⟩|H⟩), on what does the relative phase depend?
The paper is very interesting, but it again leads to my first question: It may be possible to synchronize the lasers, but how are you going to make the paths 2 and 3 *exactly* the same length? Maybe we cannot distinguish the photons by their arrival time now, but in 100 years we might have much preciser clocks that could tell the source because one path is slightly shorter than the other. Same with the orientation of the BS: You cannot *exactly* get the angle right, so if we make the paths after the beam splitter a few miles longer the beams would actually diverge and we could tell the source. It doesn’t work quite like that (as you really know since otherwise it wouldn’t work). Getting 2 & 3 to be the same length is a matter of tuning, making a slight adjustment to one path until you see the effect. Like an interferometer setup, which depends on matching as well. There is some uncertainty in the system, so you can’t expect the source times to be knowable. Ultimately, you end up with some pairs in which 2 & 3 arrive sufficiently contemporaneously to create the desired outcome.

Reference https://www.physicsforums.com/threads/why-does-a-bbo-crystal-create-entangled-photons.716479/

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