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Consider a setup in which a single photon is split into an entangled pair and directed towards 2 independent experimental setups (A and B). Both A and B have lenses to detect the polarization of the photon emitted towards them.

The photon state is described as follows where |H> is a horizontally polarized photon and |V> is a vertically polarized photon.

$|psi> = |H>_A|V>_B - |V>_A |H>_B$

This state indicates that there is a possibility that both A and B can detect a photon if the experiment is repeated enough number of times. For example, when the state of the entangled pair is $|H>_A|V>_B$ or $|V>_A |H>_B$, both setups observe a photon.

My question is, how are the energies conserved when such a state is observed ?

Since, the energy of the "parent" photon (prior to split) is given by: $๐ธ=(โ„Žโ‹…๐‘)/๐œ†$.

Are the 2 entangled photons in this state of a different color (so that the wavelength (๐œ†) is different and the energies sum up to be conserved) ?

Or, does the mechanism of the splitting the single photon necessarily add energy to the system in this case?

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First, there is conservation of energy. With a 405 nm laser driving parametric down conversion (PDC or SPDC) from a BBo crystal, the output entangled photons will center around 810 nm wavelengths. Their frequencies will total to the original input photon. (Keep in mind that most input photons do not down convert.) They are in fact entangled on this basis as well as on the polarization basis.

However, the split does not produce 2 precisely equal photons, there is some variability. This can be seen from the spray of entangled photons which emerge slightly off angle (at least with some types of crystals). They are typically collected at about 2-3 degrees off angle. There is a sweet spot for collecting the highest quality pairs for testing. That being where there is close matching of the output wavelengths.

There are a variety of crystals and setups for generating entangled pairs via down conversion. Different ones can produce different Bell states (such as the one you mention). Note that PDC can produce entanglement on the basis of wavelength/frequency without necessarily producing polarization entanglement.

Here are a few references which talk about PDC in greater detail:

https://arxiv.org/abs/quant-ph/9810003

https://xqp.physik.uni-muenchen.de/publications/files/articles_2001/journmodopt_48_1997.pdf

https://arxiv.org/abs/quant-ph/0205171

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