Is it possible to define a "it went through two slits" observable? 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.  
 A: It's a very good question but the answer is No, there is nothing such as "it went through both slits" observable (i.e. no linear operator that would correspond to this Yes/No question). The reason is that such "information" cannot be observed, not even in principle and not even statistically.
Much more generally, there don't exist any observables that would say "the physical system did X or Y before it was observed". It's the very reason why the term "observable" was chosen because things that are, by definition, unobservable because they only occur in the middle of the experiment are simply not observables.
All these statements are very manifest e.g. in Feynman's path integral formulation of quantum mechanics. One always has to sum the probability amplitudes over all conceivable histories. In this sense, using the double-slit example, the particle always goes through both slits. For some particular places where the particle may be detected, the first or second slit may contribute zero to the probability density. But the whole probability distribution is always affected by both slits.
I wrote that this question is good because these issues are misunderstood by most laymen - and not just those who admit that they are laymen but also many authors of philosophical books about quantum mechanics etc. They usually want to simplify their life and imagine that the particle, after some moment, is guaranteed to "behave as a particle". When the experimenter does something, he may forget that the particle has quantum and wave-like properties, and he may imagine it has only gone through one of the slits.
But as the "delayed choice quantum eraser" experiments show, this reasoning is always invalid. The particle may always "recall" that it has wave-like properties, and by modifying the future portions of its journey, the other slit - and the interference between both slits - may always become important once again. There's no reason why Nature should have answers to questions that can't be measured - such as "what was happening before the measurement" - and indeed, whenever we are in the quantum regime and the classical approximation is inapplicable, Nature chooses not to have answers to any of these questions.
Observables are quantities that can be shown as values at actual measuring apparatuses and quantum mechanics can only predict the odds that the value of an observable will be one number or another. Everything else that some people may want to imagine as the "detailed history" that "preceded" the measurement is unphysical.
A: The first thing I would point out is that the idea of the particle going through both slits at once is flat-out wrong. (Well I suppose you could argue various viewpoints; this is mine) It's a wave that goes through both slits; specifically, a wave in the quantum field that represents the particle, and there's no confusion about how a wave can go through both slits. Furthermore, the wave always goes through both slits - that's what waves do, they spread out in every possible direction. A bit of the wave even propagates directly through the wall!
In fact, the only reason we have this idea of particles at all is that the interaction between the quantum field of the particle and the quantum field of anything else (e.g. a detector) takes place at a specific point. Essentially, you can only have an interaction with one point on the wave, not the whole wave. When that happens, we see a localized event, so we say "oh, there's the particle." But all we're really seeing is an interaction between two quantum fields.
So when you detect a particle on the screen in this experiment, what you're really seeing is an interaction involving a point on the wave. This same wave passed through all your slits, and any other routes it could take to get to the screen. It wasn't interacting with anything along the way, so it didn't exhibit any particle-like properties while it was making its way from the emitter to the screen. Thus, it doesn't make sense to talk about which slit the particle passed through, since in the slits, there was no particle (i.e. no interaction), only a propagating wave.
A: There are observables corresponding to the light going through both slits.
You can write down a basis: "it went through slit A + slit B", and "it went through slit A - slit B". Although maybe you can't detect these observables easily with an experiment, they're perfectly good observables, they're orthogonal, and a clever enough experiment should be able to measure them (at least approximately). Something like this was done in the Afshar experiment and in the quantum eraser experiment.
But you can't simultaneously measure both the observables "it went through slit A + slit B". and "it went through only slit A" because the two observables aren't orthogonal. The observables "it went through slit A" and "it went through slit B" are orthogonal, so they form a basis. Thus, if you measure in this basis, you find it always went through one slit or the other. If you want to measure "it went through both slits" versus "it only went through one slit," this doesn't correspond to an orthogonal basis, so you can't perform that measurement.
If this were polarized light instead of slits, the measurement you want to make would correspond to "this is either vertically polarized or horizontally polarized" versus "this is right-diagonally polarized light". 
