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Is there a way to measure the entanglement of CMB photons? Could it be possible to measure such a thing and/or entanglement of photons from stars and/or cosmic neutrinos/gravitational waves and/or any other source or is that impossible with current knowledge and technology?

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You can only measure the polarization of the wave, but not photons.

Photons energy level is too small for this experiment.

bicepkeck.org/bicep2_2014_release.html

If you would like to see what they do with photons, here is an example.

The measurements use two detectors with different photon polarizer directions. When two photon sources are separated by a large angle relative to the earth, such that each detector has only one photon source in its field of view, the signal for entanglement can be separated from the background by changing the polarizer directions. When the angle between two photon sources is small enough such that both sources are in the fields of view of both detectors, then the background becomes more complicated which makes it more difficult to be subtracted.

https://arxiv.org/abs/1701.03437

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