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In the s orbital of a helium atom there are two entangled electrons. They are entangled because they have different spins.

Suppose we give energy to one electron and let it runs from the atom.

Does entanglement between electrons vanish?

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Your statement "they are entangled because they have different spins." is wrong. With different spins they could be in a combined state like this: $$ |\psi\rangle=|s\uparrow\,\rangle_1 \ |s\downarrow\,\rangle_2 $$ The extra requirement is that fermions have an antisymmetric state (which basically means they are always entangled!) In this case you would have: $$ |\psi\rangle=\sqrt\frac{\small1}{\small 2} \ \big(\ |s \uparrow\rangle_1\ \ |s \downarrow\rangle_2\ -\ |s\downarrow\rangle_1 \ \ |s \uparrow\rangle_2 \ \big) $$ If you now give one of them (say electron "1") more energy by changing its spatial orbit, but you are careful not to change its spin in the process, then you might just put it in the $p$ state for instance, giving: $$ |\psi\rangle=\sqrt\frac{\small1}{\small 2} \ \big(\ |p \uparrow\rangle_1\ \ |s \downarrow\rangle_2\ -\ |p\downarrow\rangle_1 \ \ |s \uparrow\rangle_2 \ \big) $$ or if you eject it from the atom and give it momentum $\vec k$ $$ |\psi\rangle=\sqrt\frac{\small1}{\small 2} \ \big(\ |\vec k \uparrow\rangle_1\ \ |s \downarrow\rangle_2\ -\ |\vec k \downarrow\rangle_1 \ \ |s \uparrow\rangle_2 \ \big), $$ which is in both cases just as entangled as before (in fact it is maximally entangled). So entanglement does not necessarily vanish, although it could of course change in other ways than in these examples. There may be entanglement with the environment introduced in the process.

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