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In practice, when entangling two particles (say two electrons), is creating an entangled state purely a matter of bringing them close enough together? What I mean is, to make their wave functions correlated, is it just a matter of bringing them close enough together (regardless of what is required to accomplish this), or are there other processes necessary to cause the entanglement?

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Ii may be better to say that particles become entangled by interacting. This means that some property of each particle is changed by the presence of the other. If conservation/symmetry laws impose constraints on the combined properties of the particles after the interaction, like total momentum being zero or spins being equal/opposite, but knowledge of the individual values of those properties is lost due to quantum uncertainty, then the particles are entangled. Bringing them close enough risks being a circular if the criterion for close enough is that it leads to entanglement.

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There is very good example about two entangled particles, for example We are in Lab, and we have a particle which has spin 0 ($S=0$) and that particle decays, and produces two electrons, and we have two sensors which measures spin of each electron, but according to Spin conservation sum of that electron's spins must be zero: $$ S_{electron 1}+S_{electron 2}=0\tag{1.1} $$ And for example if first sensor (detector) measures that 1st electron's spin is $\uparrow$ then second electron's spin must be $\downarrow$ because $\uparrow +\downarrow=0$ which satisfies spin conservation equation. Heres a good animation of that event:

Entanglement

Before measuring, state of electrons can be described using this equation: $$ |\psi\rangle = \frac{|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle}{\sqrt{2}}\tag{1.2} $$ (in equation 1.2 there is same probability for finding electrons in state $|\uparrow\downarrow\rangle$ and $|\downarrow\uparrow\rangle$ but after measuring wavefunction 'collapses' to either $|\uparrow\downarrow\rangle$ or $|\downarrow\uparrow\rangle$)

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    $\begingroup$ The question was how to entangle two particles, i.e. two particles initially not entangled. If you want to split a particle for producing entanglement, then your example works too hard. Use spontaneous parametric down conversion, it's the most popular technique. $\endgroup$ – Sofia Dec 8 '14 at 0:37

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