quantum mechanics - Can the absence of information provide which-way knowledge?

This seems an incredibly basic question, but one I've been unable to find an answer to on PSE; if this is a duplicate please point me in the right direction.

Concerning a simple Young's double-slit setup:

A sensor of some type is placed by one of the slits, such that if an electron were to pass through this slit, the sensor would register the passing and thus any possibility of seeing an interference pattern after many runs would be destroyed. The other slit has no such sensor.

Electrons are then fired one at a time. After each electron is detected at the downrange detection plate, a note is made whether the sensor positioned by the slit was triggered or not. In this way, two populations of detections may be built up: Marks on the downrange detection plate that were associated with the slit sensor being triggered $A$, and marks on the detection plate that had no associated triggering of the slit sensor $B$.

Now, if I observe the pattern of marks created by population $A$, I would expect to see no signs of interference as I have very clear which-way path information thanks to my sensor.

My question is this: If I choose to observe the pattern of marks created by population $B$ only, will I observe an interference pattern or not?

It seems my expectations could go both ways:

1. 
   
   

   I can argue that I should indeed observe an interference pattern since these electrons have not interacted with any other measuring device at all between the electron source and the detection plate, between which lie my double slits.

   
   


2. 
   

   I can argue that the very fact that my sensor at the one slit did not trigger a priori gives me which-way information, in that I now infer that my electron must have gone through the other slit thanks to the absence of which-way information through my sensor-equipped slit.

   
   

Which one of these assumptions aligns with reality would seem to have huge ramifications: the first implies that measurement is truly physical interaction of any kind, whereas the second implies that knowledge is measurement, even if that knowledge is obtained without physically interacting with the system (if my detector isn't triggered I cannot see how one could argue it interacted, so perhaps a more accurate statement would be there must be a different kind of interaction that may support non-epistemic views of the wavefunction).

Put another way more succinctly: It is one thing to understand that physical interaction destroys superposition. It is another to understand that a lack of interaction with a measuring device (generally pursued to preserve superposition) may also destroy it if it yields which-way information.

Given this I'm hoping the answer to my question will be #1, but expecting it to be #2.

Answer

The OP's confusion seems to stem from the incorrect assumption that

if my detector isn't triggered I cannot see how one could argue it interacted [with the electron]

Just because the detector sometimes does not click, does not mean that there is no interaction at all.

A good way to think about this is in terms of continuous measurement. This and this are good (albeit quite involved) references for further reading on this topic.

You know that, uprange of the detector, the electron probability amplitude (or if you insist, the Dirac field) is delocalised in space. In particular, there is some amplitude for the electron to be found at the position of the detector. So in fact, the detector is always interacting with the electron (continuously measuring it). However, this interaction is weak because the detector doesn't cover all of space. Therefore the electron-detector interaction is not strong enough to cause the detector to "click" (i.e. trigger it) with 100% probability on a single run of the experiment.

More precisely, at the end of the experiment the detector and the electron (or if you insist, the Dirac field) are in the entangled state (roughly speaking) $$ \lvert \psi \rangle = \lvert A\rangle_e \lvert \mathrm{click}\rangle_d + \lvert B\rangle_e \lvert \mathrm{no~click}\rangle_d,$$ where $e,d$ label the states of the $e$lectron (or if you insist, the Dirac field) and $d$etector. You can see already that there is an interaction, because the presence of the electron changes the state of the detector (which was initialised in the pure state $\lvert \mathrm{no~click}\rangle$). You run into conceptual difficulty only if you believe that the state of the detector and the electron can be described independently of each other: in QM probability amplitudes refer to the state of the system as a whole. If you do not observe the detector to click on a given run of the experiment, the state of the electron is correctly described by $\lvert B\rangle_e$. However, in order to see interference, the electron (or if you insist, the Dirac field) must instead be in the state $\lvert A\rangle_e + \lvert B\rangle_e$. Therefore there is no interference.