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If we pump a beta barium borate (BBO) crystal, we get one circle of vertically polarized photons ##\left|V\right\rangle## and an intersecting circle of horizontally polarized ones ##\left|H\right\rangle##: http://quantum.ustc.edu.cn/old/img/image002.gif

At the intersection points of the circles, we get entanglement

$$\left|\psi\right\rangle= \frac{1}{\sqrt{2}}\left(\left|H\right\rangle\left|V\right\rangle+ \left|V\right\rangle\left|H\right\rangle\right)\enspace.$$

Why do we get an entangled state and not just a mixed state

$$\rho= \frac{1}{2}\left( \left|H\right\rangle \left|V\right\rangle\left\langle H\right|\left\langle V\right|+ \left|V\right\rangle\left|H \right\rangle\left\langle V\right|\left\langle H\right|\right)\enspace?$$

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

$$\left|\psi\right\rangle= \frac{1}{\sqrt{2}}\left(\left|H\right\rangle\left|V\right\rangle- \left|V\right\rangle\left|H\right\rangle\right)\enspace,$$

on what does the relative phase depend?

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# Why does a BBO crystal create entangled photons?

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