quantum mechanics - If a photon truly goes through both slits (at the same time), then why can't we detect it at both slits (at the same time)?

I am not asking about whether the photon goes through both slits, or why. I am not asking whether the photon is delocalized as it travels in space, or why.

I have read this question:

Do we really know which slit the photon passed through in Afshar's experiment?

Which theory explains the path of a photon in Young's double-slit experiment?

Shooting a single photon through a double slit

Where John Rennie says:

The photons do not have a well defined trajectory. The diagram shows them as if they were little balls travelling along a well defined path, however the photons are delocalised and don't have a specific position or direction of motion. The photon is basically a fuzzy sphere expanding away from the source and overlapping both slits. That's why it goes through both slits. The photon position is only well defined when we interact with it and collapse its wave function. This interaction would normally be with the detector.

Lasers, Why doesn’t a photon go through the same slit every time?

Where ThePhoton says:

for example, if you put a detector after a two-slit aperture, the detector only tells you the photon got to the detector, it doesn't tell you which slit it went through to get there. And in fact there is no way to tell, nor does it even really make sense to say the photon went through one slit or the other.

In classical terms, this question might be obvious, because a classical billiard ball cannot be at two places in space at the same time. But this is not a billiard ball, this is a photon, a QM phenomenon. And this is not classical terms, but QM.

And if we truly accept that the photon travels through both slits, then it basically must exist in space at both places (both slits) at the same time.

But as soon as we interact with it (the wave function collapses), the photon becomes spatially localized, but only at a single location (at a certain time).

What is not obvious from QM, is how we can have these two things at the same time:

1. 
   

   the photon pass through both slits

   
   
   


2. 
   

   but we can only interact with it at one slit (not both)

   
   

What is that basic thing in QM, that will disallow for the photon to pass through both slits and be interacted with at both slits too? Somehow the QM world underneath will change to classical as soon as we measure, and interact with the photon. This change from QM to classical is where the possibility of the photon being at both places (both slits) at the same time gets disallowed somehow. This could be decoherence, as the QM entity gets information from the environment (because of the measurement), or just the fact that the wavefunction collapses and that has to have a single spatial location for the photon when measured.

So basically the photon goes through both slits, thus, it in some form exists at both slits at the same time. But when we try to interact with it, it will only be spatially localizable at one of the slits, not both at the same time.

Question:

1. If the photon truly goes through both slits (at the same time), then why can't we detect it at both slits (at the same time)?

Answer

If the photon truly goes through both slits (at the same time), then why can't we detect it at both slits (at the same time)?

Alright, let's play some word games:

This isn't a well-defined question. "Detect a particle" doesn't mean anything in quantum mechanics. Quantum mechanical measurements are always measurements of specific observables. There is no holistic act of "observing all properties of a system at once" like there is in classical mechanics - a measurement is always specific to the one observable it measures, and the measurement irrevocably alters the state of the system being measured.

People often use "detect a particle" as shorthand for "perform a position measurement of a particle". By definition, a measurement of position has as its outcome a single position, and interacts with the state of the particle being measured such that it now really is in the state in which it is at that single position and nowhere else. So if you could perform position measurements that yielded both slits as the position of the particle, this would mean you have performed an impossible feat - there are now two particles, each in the state of being at one slit and that slit only. Quantum mechanics may be weird, but it is hopefully clear it is not this weird - we cannot duplicate a particle out of thin air just by measuring it.

If you don't insist on "detect" meaning "performing a position measurement", then of course the standard double slit setup is a "detection" of the photon at both slits - the pattern on the screen is only explainable by the particle's wavefunction passing through both slits and interfering with itself. This is of course just indirect reasoning - there simply is no observable whose eigenstates would naively correspond to "we have detected the photon at both slits at once".

Lastly, you seem to confuse "interacting" with "measuring" or "detecting". Of course we can interact with the particle at both slits - we just cannot perform position measurements (or other "which-way" measurements) at both slits and expect them to yield the impossible result of the particle split in two. But if you look at more sophisticated setups like the quantum erasers, there certainly is interaction with the particle at both slits - just carefully set up to not destroy the interference pattern, and hence no obtaining useable which-way information.