I'm trying to read the "entanglement between photons that never coexisted " from 2012. Avaliable at: https://arxiv.org/abs/1209.4191

And there's this equation:

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|φ± \rangle = \frac 1{√2}(|HaHb ± |VaVb\rangle)

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|ψ± \rangle = \frac 1{√2}(|HaVb ± |VaHb\rangle)

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where ha(vb) represents a horizontally (vertically) polarized photon in spatial mode a (b).

Soooo, I guess what it is saying is:

When photons are entangled, their polarization can be entangled in 4 different ways:

φ+ they have the same "polarization degree" (like vertical and vertical, or horizontal and horizontal) and the same phase (like when one wave is in "Crest" the other will also be in "crest")

φ- they have the same "polarization degree" but opposite phases.

ψ+ they have the opposite "polarization degree" (like horizontal and vertical) and the same phase.

ψ- they have the opposite "polarization degree" and the opposite phase.

And in our BBO cristal we create entangled photons, but we have no clue on what kind of entanglement they have until we measure it.

So far I understood correctly?

Thanks! =)