# Problem with the polarization of entangled photons

• I
• Christian Thom
Christian Thom
Consider this thought experiment : we use a source of identically vertically polarized photons, such as produced by a type 0 SPDC. One beam go to Alice and the second to Bob.
1. Whatever measurement Bob makes on its beam, if Alice use a vertically polarized detector, all photons are detected, due to the nature of the source.
2. If Bob use a detector with a polarization @ 45°, about half of the photons are detected.
3. Now Alice places its detector @ 45 ° too. The twin photons of those who are detected at Bob's are also detected at Alice's, since they are entangled, but they would have been also detected with a vertically polarized detector, as seen in 1.

So here is the problem : what is the polarization of these photons since they pass at 100 % in two differently oriented polarizers ? Please point out where is my mistake.

Christian Thom said:
Consider this thought experiment : we use a source of identically vertically polarized photons, such as produced by a type 0 SPDC.
Pairs of identically polarized photons are not in an entangled state.

but in type-0 or type-I SPDC you have
$$|\Psi \rangle=\frac{1}{\sqrt{2}}(|H_i H_s \rangle+ \exp(\mathrm{i} \phi) |V_i V_s \rangle)$$
which are entangled states.

DrClaude
Christian Thom said:
vertically polarized photons
If they are vertically polarized, then ##|\Psi \rangle=|V_i V_s \rangle##, which is not entangled.

DrClaude
Ok, in type-0 SPDC this you get if the pump photon is in the state ##|V_p \rangle##, and this doesn't produce entangled two-photon state. If you use a ##45^{\circ}##-polarized pump photon you get the entangled state given #3.

kered rettop and DrClaude
Christian Thom said:
Consider this thought experiment : we use a source of identically vertically polarized photons, such as produced by a type 0 SPDC. One beam go to Alice and the second to Bob.
1. Whatever measurement Bob makes on its beam, if Alice use a vertically polarized detector, all photons are detected, due to the nature of the source.
2. If Bob use a detector with a polarization @ 45°, about half of the photons are detected.
3. Now Alice places its detector @ 45 ° too. The twin photons of those who are detected at Bob's are also detected at Alice's, since they are entangled, but they would have been also detected with a vertically polarized detector, as seen in 1.

So here is the problem : what is the polarization of these photons since they pass at 100 % in two differently oriented polarizers ? Please point out where is my mistake.
Just to add to the correct answers above by @DrClaude, @vanhees71 , @Hill :

It is possible to generate vertically polarized photon pairs using SPDC. They are entangled in some bases, but they won’t be polarization entangled.

Measuring both of them at 45 degrees- your #3 - will show 0 correlation. They show 100% correlation only when measured as V.

Hill
Thank you all for your answers that clarifie the situation. I guess the impossibility to have this kind of entanglement is not limited to spdc, but generalizes to other methods of producing entangled pairs.

It's all about the polarization state of the pump photon relative to the axis of the birefringent crystal used for SPDC. Let's denote the ordinary axis of the crystal as the ##x##-axis (and the polarization state with ##|H \rangle##) and the extraordinary axis as the ##y##-axis (and the polarization state with ##|V \rangle##). In type-0 SPDC a H-polarized pump photon splits in two photons ("idler and signal photon") in the state ##|H_i H_s \rangle## and a V-polarized one in ##\exp(\mathrm{i} \varphi) |V_i V_s \rangle##. Here ##\varphi## is the phase difference between the one or the other case.

For an arbitrarily polarized pump photon with state ##\alpha |H \rangle + \beta V \rangle## you get ##\alpha |H_i H_s \rangle + \beta \exp(\mathrm{i} \varphi) |V_i V_s \rangle##. Of course ##|\alpha|^2+|\beta|^2=1##, and thus you get an entangled state if neither ##|\alpha|^2=1## nor ##|\beta|^2=1##.

kered rettop

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