Say it's a normal double slit experiment, and the particle is detected on some position on a wall after the slits. How much info could be deduced from the position?
I know it's not exactly a path per se in a classical sense, but, based on the position of detection on the wall, how much info could be known about how the particle interferes with itself crossing the double slits?
This might be better answered with a different situation. A distant star emits light in all directions. As part of all this, a photon is emitted. What path is it on?
There is no way to tell while it is in flight. It spreads out in all directions like a wave. It might wind up anywhere.
Eventually it hits your eye. Now you know where it wound up. Can you say it followed a straight path?
You can collect a lot of photons that enter your eye. You can try things out. Put an object between your eye and the star. It blocks all photons. Put it somewhere else. It doesn't block photons from the star. You conclude that photons travels in straight lines and hits a receptor in your eye like a particle.
Try some more. Put a wall between you and the star. Put a hole in it. If the hole is big, you get particle-like results as before. If you shrink the hole, you find that some of the photons miss your eye and hit a receptor in the eye of a person next to you.
Now they are acting a little like a particle and a little like a wave. You know they wound up at the hole, but you can't say where in the hole. You might think that limits any path they took. They don't hit the wall, but continuing that path doesn't predict which eye they hit. You need to use waves to make a prediction of where they might hit. But when they do hit, they only hit one receptor in one eye.
Put two slits in the wall, so that you are between them and neither aligns with the star. Some photons hit you. None hit the person next to you. Some hit the next person over.
Photons don't change back and forth between a particle and a wave. They are kind of like both. See How can a red light photon be different from a blue light photon?. Sometimes a particle following a path is a good approximation to what they do, and sometimes not.
Ray optics treats them like straight line particles. It works very well, but not perfectly. When light from a distant star hits a camera, it passes through a large hole. The lens focuses all rays onto a single receptor. But passing through a hole introduces a tiny uncertainty in which receptor will be hit. This is called diffraction. It is the wave like nature of light limiting how well ray optics works. A perfectly designed and manufactured lens is said to be diffraction limited.
So sometimes light travels in a straight line, sometimes almost in a straight line, and sometimes not much at all like a straight line. You can derive all of these behaviors from wave properties of light. See Explanation of diffraction of a single light ray by Huygens' principle.
This tells you that light has a spread out nature. You have to add up contributions from everywhere that light is to find out where it will be next. If light is widely spread out, like a plane wave from a distant star, the prediction is that it will continue in the same direction.
Then light hits a hole in a wall. Now it fills the hole, but no farther. If you add up just that much, the wave largely continues straight but spreads out. For a small hole, it spreads out a lot.
But this wave is not a classical wave. It doesn't tell you where the light is. It tells you where you might find light if you measure it. This spread out wave can hit a single receptor. It is something like a particle.
Also you should not think of the particle like nature as a classical particle. In particular, it does not have a size. It can fit a hole or a receptor.
The important particle like property is that the energy of light comes in lumps. When a photon hits a receptor, one lump of energy hits. Nearby receptors get no energy. When a photon passes through a hole or double slits, either all the energy passes through, or none.
The wave tells you where the energy is likely to wind up, but not which receptor will be hit.
This is one of the central counter intuitive parts of quantum mechanics. It is so unlike the cause and effect we are used to from classical physics, that we struggle to make sense out of it. The immediate question is "If a spread out wave arrives at a bunch of sensors, what causes it to pick just one?"
The answer is that cause and effect does not work this way. You just have to get used to it. Nothing travels from distant points of the wave to the lucky receptor. Nothing travels from the lucky receptor to other receptors. It just turns out that one receptor is lucky and all the others are not. All cause and effect has to say is the if a photon arrives, it will arrive somewhere. The wave tells you the probability of where.
...how much info could be known about how the particle interferes with itself crossing the double slits?
A useful version of this setup is to consider one in which individual photons are passing through a double slit on their way to a screen (or other detector type). Because the photons go through one at a time, any interference pattern on the screen will be built up strictly as a result of self-interference. Self-interference only occurs when there are 2 (or more) indistinguishable paths from the source to the screen, and at least one of those paths go through each slit. You probably know this much already.
It is possible to estimate the odds of the light hitting spots on the screen. Usually, that is done by considering the pattern of bars that result. The pattern of course changes when interference is possible versus when interference is not possible (such as when one slit is blocked).
There is only interference to the extent that there is indistinguishability between photon paths going through the Left slit, versus going through the Right slit. So to answer your question:
In principle: No "which-slit" information (i.e. information about self-interference) can be gained at all from the location of any individual photon's mark on the screen. Otherwise there would be distinguishability, and then there is no interference. You just get a single crested pattern on the screen, which is the indication that you could (in principle at least) determine the photon path through one slit or the other.
In practice: Being able to distinguish the path through one slit or the other is not an all-or-nothing proposition. You could, for example, have it such that you are 60% sure the photon went through the Left slit, and 40% sure the photon went through the Right slit. In such cases, the patterns become more blurry. There are specific experiments in which this has been accomplished (see second reference below). Further, no experiment is ideal, so there are always some photons that are distinguishable regardless. The resulting pattern is never perfect.
Young’s Double-Slit Interference Demonstration with Single Photons (2024) Note: This paper covers both theory and experiment, and is relatively straight-forward and fully up-to-date.
"The interference of single photons going through a double slit is a compelling demonstration of the wave and particle nature of light in the same experiment."
Young's double-slit experiment with single photons and quantum eraser (2012) Note: this is a more advanced version of the double slit, but a lot of good stuff is presented.
"An apparatus for a double-slit interference experiment in the single-photon regime is described. The apparatus includes a which-path marker that destroys the interference as well as a quantum eraser that restores it."
