How does particle indistinguishability reconcile with the quantum Zeno effect? In  non-relativistic quantum mechanics, two particles are said to be indistinguishable if the wavefunction for the system consisting of just those two particles satisfies $|\psi(x_{1}, x_{2}, t)|^{2} = |\psi(x_{2}, x_{1}, t)|^{2}$ for all time $t$. (And of course, any two fermions or two bosons of the same particle type are known to be indistinguishable.)
I'm a bit confused how this can be reconciled with the notion of the quantum Zeno effect, which is the fact that when the same measurement is done in quick succession repeatedly, the evolution of the system being measured tends to slow down, and the system essentially freezes in the limit of making more and more measurement in a given time span. This makes sense given how projective measurements work, and this has been experimentally observed.
However, this seems like it could be used as a way to distinguish particles, which shouldn't be possible.
Suppose I have two electrons, each in a separate ion trap, and now suppose I measure the positions of the two electrons at a time $t$.
Now if I measure the positions of the two electrons at a time $t+\epsilon$, shouldn't it be the case that the electron measured in ion trap 1 at time $t$ is the same electron measured in ion trap at time $t+\epsilon$? Why is it invalid to conclude this?
If the two electrons are stored in ion traps such that they can't escape or travel across the laboratory, and if we continue to do repeated measurements such that their joint state doesn't evolve, shouldn't it be possible to say that the electron inside ion trap 1 continues to be inside ion trap 1?
 A: First of all, note that you could apply the same argument to two electrons bound to two far-apart atoms, without invoking any measurement. You can be reasonably sure that the particles aren't switching places if they're on opposite ends of the galaxy! So it's perfectly reasonable to say that you have the same electrons at each position at any given time, assuming that the notion of 'same electron' is a meaningful concept in the first place...
In classical mechanics, the notion of 'same particle' makes pretty clear sense, and quantum mechanics reproduces classical mechanics in certain limits. So it makes sense that some notion of 'same particle' would manifest at least in situations where quantum effects are negligible, such as in the above example (where the particle spacing is much larger than the quantum uncertainty in position). In contrast, quantum mechanics gives starkly different predictions when wave functions overlap, because one sees interference. A pretty good analogy (that one takes quite seriously in quantum field theory) is that identical particles are ripples in the same medium, and therefore can interfere. If you have two closely-spaced ripples, who's to say which ripple is which? Is that even a meaningful question? Conversely, distinguishable particles can't interfere any more than a water wave can interfere with a radio wave.
One last point that may help to clear up confusion is about your definition of distinguishability. This symmetry of the wave function in your question will certainly be true for distinguishable particles, but it is not the definition. Particles are indistinguishable if there is no measurement or experiment that can distinguish them even in principle. For instance, if it turned out that electrons had some undiscovered property that current experiments couldn't detect (maybe some are happy and some are sad), one might have $|\psi(x_1,x_2,t)|^2 = |\psi(x_2,x_1,t)|^2$ in all current experiments, but future detectors might be able to distinguish them. However quantum mechanical interference between electrons implies that no such additional properties exist (or perhaps it's just that all electrons are happy :).
