Do quantum measurements destroy the symmetrization requirement?

Question

Do quantum measurements destroy the symmetrization requirement? The symmetrization requirement requires that the combined wave function of a system of particles be either symmetric under an exchange of two particles (bosons) or antisymmetric (fermions). For instance, if you have two spin 1/2 fermions in the singlet spin configuration, then (assuming the two particles are in the same orbitals), the (total) wave function is antisymmetric (as required). But if you actually go and measure their spins, then the wave function collapses into a state with one particle having spin up and the other spin down. Now the (spin space) wave function is no longer antisymmetric, and hence the total (spatial + spin) wave function isn't either. In general, the symmetrization requirement means that two particles must be entangled in some way (meaning you can't write the total wave function as a simple product of single-particle functions (so long as you're not talking about a bunch of bosons in the same state)), but this entanglement is usually destroyed under measurement. So what's going on here?

Answer 1

Indeed this is a subtle point. The key point is that a measurement doesn't tell you which electron has spin up, just that one of them does. A second important fact is that only the total state needs to be anti-symmetric.

Let's take an example with two electrons. We will define the state $|x \uparrow, y \uparrow \rangle$ to mean electron $1$ is at position $x$ with spin up along some axis, and electron $2$ is at position $y$ with spin up along the same axis. Let's say they are in an anti-symmetrized state like so \begin{equation} |\Psi\rangle = \frac{1}{2}\left( |x\uparrow, y \uparrow\rangle + |x \downarrow, y\downarrow\rangle - |y \uparrow, x \uparrow \rangle - |y \downarrow, x \downarrow \rangle \right) \end{equation} In this state, neither electron is in a definite spin state.

Now we do a measurement, and find an electron at position $x$ is in the spin up state. You might think this means the state after measurement becomes \begin{equation} |\Psi\rangle = |x \uparrow, y\uparrow \rangle \ \ {\rm (NOT\ CORRECT)} \end{equation} In fact this assumes that we have measured the first electron to be at position $x$ with spin up, but since electrons are indistinguishable there is no way to be sure whether we have measured the first or second electron. Therefore the correct state after measurement is \begin{equation} |\Psi\rangle = \frac{1}{\sqrt{2}} \left(|x \uparrow, y\uparrow \rangle - |y \uparrow, x\uparrow \rangle \right) \end{equation} which is anti-symmetric, and has been projected to the subspace where one of the electrons is at position $x$ with spin up (consistent with our measurement). This logic generalizes fairly straightforwardly to systems with $N$ identical fermions.