Are particles entangled after beta decay? Are the proton, electron and anti-neutrino entangled after beta decay?
If you measured the spin of the neutron before beta decay and it was $-1/2$ and you measure the spin of the electron afterwards and it is $+1/2$ then you would instantly know the spin of the proton and the anti neutrino. But if the electron's spin is $-1/2$ you wouldn't know the spin of the proton and the anti neutrino. So are they entangled or not?
 A: The entanglement is with spin and it a tripartite entanglement. Suppose we have a neutron with the spin state
$$
|\psi\rangle~=~\frac{1}{\sqrt{2}}|n\rangle\left(|+\rangle~+~|-\rangle\right).
$$
The $|\pm\rangle$ states correspond to the spin up and down. The neutron state decays into anti-neutrino and and proton states. The state is then
$$
|\psi\rangle~\rightarrow~\frac{1}{2}|p\rangle|\bar\nu\rangle|e\rangle\Big[|+\rangle_p(|+\rangle_e|-\rangle_\bar\nu~+~|-\rangle_e|+\rangle_\bar\nu)~+~|-\rangle_p(|+\rangle_e|-\rangle_\bar\nu~+~|-\rangle_e|+\rangle_\bar\nu)\Big]
$$
$$
=~\frac{1}{2}|p\rangle|\bar\nu\rangle|e\rangle\Big[\big(|+\rangle_p|+\rangle_e|-\rangle_\bar\nu~+~|+\rangle_p|-\rangle_e|+\rangle_\bar\nu\big)~+~\big(|-\rangle_p|+\rangle_e|-\rangle_\bar\nu~+~|-\rangle_p|-\rangle_e|+\rangle_\bar\nu\big)\Big].
$$
This is a superposition of two entangled states. Here we have that the particle states $|p\rangle$, $|e\rangle$ and $|\bar\nu\rangle$ correspond to other quantum numbers, such as baryon, lepton and antilepton numbers and these do not enter into the entanglement.
