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My question is whether or not it would be possible to create an entangled state between two photons that do not share the same initial photon source and their respective sources are separated by an arbitrary distance in space. This is a curiosity of mine and I have a very basic understanding of this phenomena at present, though I am attempting to learn more. If it is the case that this is possible, I would also appreciate an explanation of why or at least have someone point me to a resource explaining this. Thank you.

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If the two photons are the only systems you are considering, the short answer is no, you cannot entangle them by performing only local operations. This is a consequence of the fact that entanglement cannot increase under Local Operations and Classical Communications (LOCC) only. See http://en.wikipedia.org/wiki/LOCC for a reference.

However, as it is pointed out in the previous answer, if the photons were allowed to interact with other systems, then the answer is yes, it is possible to entangle them. One way to do this is the following: Suppose Bob in one location is in possession of one of the photons while Alice is in possession of the other, the combined system being unentangled. Locally, Bob can prepare a maximally entangled state of his photon and an additional he has with him. He then sends the latter photon to Alice, so that now they both share an entangled pair. Note that it is an entangled pair where Alice holds a different photon than the one she had originally.

If Alice wants to "transfer" this entanglement to her original photon, she can also entangle it with an additional photon she keeps locally, and then perform a Bell measurement on the photon sent by Bob and the new one she introduced. Depending on her outcome, she performs a local operation on her original photon and in the end this will be entangled with the one Bob has in his location.

What I explained in words is referred to as entanglement swapping, an operation based on quantum teleportation. For more details you can take a look at http://en.wikipedia.org/wiki/Quantum_teleportation#Entanglement_swapping.

Of course in practice, entangling two photons is bloody difficult, but I presume you were interested on whether it could be done in principle and if so, under what conditions. Also, these operations are possible in principle regardless of what physical system is used.

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Thanks alot to both Terry and Juan. You have both been very helpful. I understand now how this possible and to be honest I had a tentative notion that it would be something like what Juan has described here as entanglement swapping. Thanks again –  user10120 Jun 27 '12 at 23:50
    
No problem! Keep asking interesting questions! –  Juan Miguel Arrazola Jun 28 '12 at 2:47
    
As I understand, the entangled state is destroyed if either photon of the entangled pair is measured. Since entanglement has some utility in cryptography and communication, this is necessarily a resource that must be replenished. So, after one round of entanglement swapping and subsequent destruction of the entanglement (for its utility), do Alice, Bob, and Carol have to then meet up and reset to the previous state of affairs? What I am looking for is a way to preserve a sort of "master" entanglement that is not destroyed that will serve to generate new entanglement (for utility) remotely –  user10120 Jun 28 '12 at 5:36
    
Any local measurement on a bipartite entangled state will necessarily have the property that the conditional states for each possible outcome of the measurement are product states. So if Alice and Bob each make a local measurement, whether the physical system is "destroyed" or not, they will be left with a product state. Perhaps you can show this yourself? –  Juan Miguel Arrazola Jun 29 '12 at 20:06
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The answer is yes, you can entangle two photons that do not share a common source.

The trick is that that both of the photons from separate sources must be entangled with particles from those two sources. These other particles must then "interact" in some way that constrains the properties of the two photons you want to entangle.

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Thanks Terry, Iappreciate your insight. I still have to ask, however, if it is possible for two separate sites to acheive entanglement of a pair of photons without ever having two convene with one another except through entirely classical means like LOCC. What I mean is can this be acheived without resorting to quantum mechanical operations or any a priori entanglement. How would the two separate sites acheive this? –  user10120 Jun 27 '12 at 22:31
    
I'm not fully sure if I'm reading your question correctly, but if you are asking whether entanglement can be achieved using only fully classical operations to connect them in spacetime, the answer is pretty easy: no. Quantum uncertainty and classical uncertainty are mutually exclusive to the point of inversely defining each other. Here's the easiest way I can think of explaining why what you asked is not possible: If the link between two events if fully classical, it means that chain has no uncertainty left. Any non-classical results at either end thus would generate a causality paradox. –  Terry Bollinger Jun 30 '12 at 23:24
    
Thanks Terry. That really clears up my thinking! –  user10120 Jul 2 '12 at 3:33
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