# Can different components of a wavefunction become entangled with different systems?

Imagine a $z$-basis spin Stern Gerlach experiment where a single "particle" is put through the apparatus.

Upon exiting the Stern Gerlach magnets, the result is the sum of two states. One state is the spin-up state, multiplied the probability of being in that state and also multiplied by a $z$-position Gaussian packet (over one range of $z$ states). The other state is the spin-down state, multiplied the probability of being in that state and also multiplied by a $z$-position Gaussian packet (over a different range of $z$ states). These two Gaussian packets are moving in opposite $z$-directions, and not overlapping in space (not overlapping in a significant way).

\begin{align} \Psi' &= \bigl(e^{-({z - z_\text{up})^2}\Delta}\bigr)\alpha \lvert\text{up}\rangle + \bigl(e^{-({z - z_\text{down})^2}\Delta}\bigr)\beta \lvert\text{down}\rangle \\ &= \Psi_\text{up}' \lvert\text{up}\rangle + \Psi_\text{down}' \lvert\text{down}\rangle \end{align}

There is also a time dependence factor that is suppressed ($z_\text{up}$, $\Delta$, and $z_\text{down}$ are changing over time).

Then, at $t = t_1$, is it possible that $\Psi_\text{up}'\lvert\text{up}\rangle$ could subsequently become entangled with some other (hypothetical) system ("system A"), while $\Psi_\text{down}'\lvert\text{down}\rangle$ does not, but instead becomes entangled with some other system B? Here, A and B are located in different places along the z axis.

If so, system A and system B each evolve, at least partly, due to the Gaussian packet it interacted with. Assume system A and system B are not directly measured, for the purpose of this discussion.

Suppose, at $t_2 > t_1$, the particle's spin is then measured and the result is spin up. In that case, would that part of the evolution of system B, between $t_1$ and $t_2$, due to the particle's $\Psi_\text{down}'\lvert\text{down}\rangle$ that it interacted with, become "nullified"(or maybe, more technically, "post-selected out"?

Another way to look at this is from a decoherence perspective. Suppose $\Psi'$ is the system and A and B are objects of the "environment". I think decoherence theory posits that $\Psi_\text{up}'$ could become entangled (correlated) with a state of A, while $\Psi_\text{down}'$ could become entangled (correlated) with a state of B (if A is within the $\Psi_\text{up}'$ Gaussian position packet and B is within the $\Psi_\text{down}'$ Gaussian position packet). But $\Psi_\text{up}'$ does not become correlated with a state of B and $\Psi_\text{down}'$ does not become correlated with a state of A. (By "within", I mean within a few standard deviations of the center of the Gaussian.) So, I am thinking that it is possible for two different components of the same wave function (which add together to produce that wave function) can entangle with states of two other, different, systems.

• I'm pretty sure this is not correct, but I'd still like to clean up the original question before answering. For example, I still think having both the momentum and position wavefunctions is unnecessary. It's also confusing because some of the important bits of the Fourier transform are hidden in $\alpha$ and $\beta$. Any chance you could simplify the original post, perhaps starting with the position wavefunctions? Aug 31 '16 at 17:39