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Suppose we'd like to know whether two cosmic microwave background photons emitted from different parts of the sky have any quantum correlation with each other. We could measure polarization of two photons in different directions and see if results of the measurements violate Bell's inequality. Thus we could characterize degree of entanglement between photons emitted from two distant sources.

Is this a realistic experiment and has it already been done?

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    $\begingroup$ The polarization of the CMB is being measured with the BICEP2 experiment, but it measures the polarization of the classical wave, a collective phenomenon, not individual photons bicepkeck.org/bicep2_2014_release.html . I do not think an experiment is possible on individual photons, their energy is very small so cannot excite specific lines in atoms or intract with electrons that could be measured in an aparatus. $\endgroup$
    – anna v
    Commented May 30, 2015 at 13:18

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[Note: This question is old, but I am answering as it was referenced in a similar question from 12/12/2023. It had no answers, and was closed as of the time I wrote this.]

The simple answer is that Cosmic Microwave Background Radiation (CMBR or CMB) photons will never violate a Bell Inequality. There are a couple of reasons for this:

a) Suppose (some or all) CMB photons were entangled. No pair can be close enough to being maximally entangled, which is a requirement for violating a Bell inequality. There is a principle in QM called Monogamy of Entanglement (MoE). It says that if you have any photon pairs AB and BC: If AB is maximally entangled, AC cannot be entangled at all. There is always a trade-off: AB can be partially entangled and BC can be partially entangled too. But that is not going to be enough to violate a Bell inequality. It might be enough to detect some correlations between 2 of any N=3 photons, but would not be enough for any randomly selected photons from the CMB (where N is a very large number).

From Wiki: https://en.wikipedia.org/wiki/Monogamy_of_entanglement

b) More importantly: How would any CMB photons become entangled in the first place? Such photons were created during an early epoch (approx. 380,000 years) of the then hot universe and then were scattered. There was no particular relationship between the various sources of those photons. The photons later cooled by a factor of about 1000 down to the current 2.7 degrees (approx.) as the universe expanded. In the subsequent time period, those photons would not have had anything happen to them to entangle them. Even if they had originally had some similar polarization, that is not entanglement (for example lasers emit photons that have the same polarization but are not entangled).

It might be possible to corral pairs of individual photons from the CMB, but they would demonstrate Bell type correlations. For example, they would likely be from different parts of the sky and would have no particular connection other than their old age.

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