To test particle entanglement at a distance, do they have to start in proximity or can they be identified already distant? If so, how?
As in Holger Felder's Answer, to make an entangled state, all known ewxperimental techniques are local insofar that entangled states must be 'created' by a single interaction: you need to produce a pure quantum state e.g. of two photons with opposite spin in a given direction, so that the pure quantum state is nonfactorisable (i.e. can't be written in the form $\left.\left|\psi_1\otimes \psi_2\right.\right>$, where $\left.\left|\psi_1\right.\right>$ and $\left.\left|\psi_2\right.\right>$ are independent one-photon states).
Even in theory, two spatially separated particles can become entangled, but only by a special communication protocol between the two particles' locations called Entanglement Swapping (see Wiki page "Quantum Teleportation" and the "Entanglement Swapping" section. The entanglement must be "transmitted" from one location to another.
But I think you're asking whether one can tell whether two particles are entangled when they have been entangled elsewhere and travelled to the experimenter's location. There are two important things to heed here:
- Any entanglement experiment is like this, even if the detection apparatus is only a few centimetres away from where the entangled particles are produced. So you can in theory detect entanglement arising from photon productions in the next galaxy just as well as you can detect entanglement arising from their production in the next room, as long as you can still access the pairs. In practice, you would be unlikely to know which pairs of photons are meant to be entangled if they just dropped in for afternoon tea in your laboratory casually after having made a journey from M87;
- NO form of entanglement can be confirmed by the observation of only ONE pair of particles. Quantum mechanical experiments are probabilistic in nature, so the only thing you can do is make many measurements and confirm whether or not the correlations between the measured states is statistically significantly higher than the limits laid down by the Bell Inequality for non-entangled particles.
A kind of exception to point 2. is if you measured the state of one photon and confirmed it to have collapsed to, say, left circular polarisation and measured the other also to have collapsed to the same polarisation then you can say that they were, to within your experimental accuracy, not in the entangled state as follows:
So you could conclude, even after one observation, that they probably were not produced in a single interaction producing photons of opposite spin.
But note that this STILL does not rule out entanglement. There is nothing to say that the photons were not in the following nonfactorisable state before the measurement:
even though I don't know of any experimental apparatus that could make such an entangled state.
See this article: Časlav Brukner, Markus Aspelmeyer, and Anton Zeilinger, "Complementarity and Information in "Delayed-choice for entanglement swapping", arxiv.org/abs/quant-ph/0405036v1 . You can find it in Internet, in the arXiv quant-ph. By the way, it is BETTER to read the article than the issue in Wikipedia about entanglement swapping. The treatment in Wikipedia is more complicated than needed.
The idea is as follows: two identical pairs of photons are produced by identical sources (e.g. by down-conversion in identical non-linear crystals illuminated with identical ultraviolet beams). In such pairs, one of the photons is called signal, the other idler.
From each pair, one photon, say, the idler, is sent to a common observer, Victor, that projects the pair on a certain Bell-type state. A Bell-type state is an entangled state, and there are four such states - see in the article formulas (3) - (6). The signal photons were sent, one to an observer Alice, and one to an observer Bob, these two observers being far from one another.
Well, in fact, Alice and Bob performed their measurements, each one on her/his signal-photon, before Victor performed his measurement on the two idlers. So, Alice made her measurement and Bob too, then Victor. But when all three measurements were compared, it was found that Alice's and Bob's particles were entangled in the same Bell-type state as Victor's idlers.
One could suggest that Victor's measurement collapsed the two pairs of photons, imposing thereby an entanglement also between Alice's and Bob's photons. But Victor measured after Alice and Bob. The facts go as you asked, by the time that Alice and Bob measured their particles, these should have been independent, not entangled.
Good luck !
P.S. Just keep in mind some issue. Entangled particles seem to ignore space and time. I say "seem" because we know very little on entanglement, neither can we say what is the wave-function. We are only able to play with formulas. Thus, about the fact that Victor makes his measurement after Alice and Bob, doesn't mean much for these particles. Moreover, we can find a moving frame of coordinates in which Victor measures first, and Alice and Bob later.
To make two photons entangled they had to be produced together. For example with down-conversion in non-linear crystals or from quantum dots which emit pairs of photons. This is needed for quantum cryptography. On the other side the photons from a laser beam if one let them through a polarisator are entangled in their frequency and in their field direction. But this is not bugproof because one can't count the missing photons and the misalignment of the system from the spy's influence.
The strange thing is the point of view that the artfully made quantum dot does not produce the photons in the state we measure them but in mixed states. In mathematical expressions this is correct and describes our knowledge in the time between the pair production and the measurement. The oposite point of view that the photons are produces in the states we measure them is not more or less to proof then the mixed states. This is the same as one don't know is the tree falling in the forest until he is not going to proof it.