This concerns the famous two-slit experiment.  Electrons or photons or your favorite particle, doesn't matter.   As we all know, the attempt to detect which slit the quanta pass through leads to loss of the diffraction pattern.  

The interesting part of the experiment is describing what happens when neither slit has a particle detector.  It is commonly said that each particle goes through both slits at the same time.  At the popular level this makes QM seem weird and mystical, and at the professional level we might cite the Copenhagen Interpretation which boils down to: Don't ask questions about things you can't observe, and so we don't speak of such things, just do the math.

How sound this reasoning is at a fundamental level? Can we really conclude each quantum goes through both slits yet as a whole entity?   Can we define a "it went through both slits" observable?  Is there a proper Hermitian operator with (I suppose) eigenvalues 0 and 1, that can distinguish a quantum going through both slits vs. only one slit but without saying which one?

Perhaps it would make more sense to think about an N-slit experiment, and ask about a Hermitian operator that can report n, the number of slits a quantum takes?

I suspect I'm not asking this question quite right, but have an intuition there's something yet to be dug up from this age old gedankenexperiment.