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Quantum teleportation and super dense coding use entanglement. You can teleport a qubit, storing A LOT of information in the coefficients $\alpha \vert0> + \beta \vert 1>$ by communication only 2 classical bits of information.

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What you've described seems to be one example. In general, interactions between two quantum systems will put the system into some joint state which will generically be entangled. For example, if you have two spins coupled with a spin-spin coupling, then the ground state of that (total) two spin system has some entanglement.

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So, out of lack an already established name, I called it excitation states in: P. Migdał, J. Rodriguez-Laguna, M. Lewenstein, Entanglement classes of permutation-symmetric qudit states: symmetric operations suffice, arXiv:1305.1506.

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Ignoring the fact that this is a reaction with very low probability and the difficulty of detecting such an interaction the answer is : no, nothing happens to the quantum mechanical status of the other gamma. If by some ingenious method one managed to determine the spins in the pair produced, still the spin of the parent gamma would be unknown, since the ...

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We have to be careful with the bra-ket formalism and its meaning. Unlike $|x_1>$, I am not sure that the notation $|x_1 x_2>$ where $x_1$ and $x_2$ are positional coordinates makes any sense. In literature [1] the notation $|ab>$ designates the Slater determinant or Hartree-Fock state, i.e.: |ab>=c_a^{\dagger}c_b^{\dagger}|0>=\phi_a(x_1) ...

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This question actually has a very easy and rigorous answer. Having which-path information "available" is just a crude way of saying that the system is correlated with anything else. Usually, this is because the system has been decohered in whatever basis corresponds to the possible paths, which is usually position basis. In your case, the photon is ...

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The question asked is relevant and something that is usually ignored because no-one understands the EPR paradox. (well I have ideas: http://quantummechanics.mchmultimedia.com/ I find that when students come up against this question of non-locality (persistence of entanglement) they are puzzled and, like good scientists, ask their mentors. The mentors ...

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