# 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.

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.

• It is not entirely clear from the way you write it, but it seems that you say that a $|+\rangle_n$ always gives rise to a $|+\rangle_p$. For sure, in the state you write the proton is unentangled with the electron and neutrino. Is there a reason for that? – Norbert Schuch Sep 21 '17 at 13:36