# Does an entangled pair remain entangled after the first measurement?

Consider an entangled pair described by the wavefunction $$\lvert1,0\rangle = \frac{1}{\sqrt{2}}(\lvert\uparrow_1\downarrow_2\rangle-\lvert\downarrow_1\uparrow_2\rangle)$$ in in the $$S_z$$-basis. If the first measurement finds Alice's particle to be in the $$\lvert\uparrow\rangle$$ state, then Bob's particle is found to be in the $$\lvert\downarrow\rangle$$ state i.e. the entangled state above collapses to $$\lvert\uparrow_1\downarrow_2\rangle$$ which is not entangled. Since it is not entangled in the $$S_z$$ basis, it will also be unentangled in the $$S_x$$ basis.

Am I correct to conclude from this that an entangled pair becomes unentangled after the first measurement? Here, I am assuming that making a measurement of $$S_z$$ by Alice or Bob doesn't affect the space part of the wavefunction, which I am not sure though.

• A comment about your last line - Two particles can be doubly entangled i.e. entangled in spin and some other property like position. In such a case, measuring the spin will not destroy the entanglement in the other property.
– rnva
Commented Oct 28, 2020 at 12:28

If you do a projective measurement with respect to a basis made of separable states, then yes, every post-measurement result will be separable. In your example, you are essentially considering a measurement wrt the basis $$\{|s,s'\rangle : \, s,s'\in\{\uparrow,\downarrow\}\}$$ (that is, the basis of eigenstates of $$S_z\otimes S_z$$), therefore the measurement breaks the entanglement.