the experiment allows you to find out "whether the bomb is able to explode" without "exploding it".There is clearly no real paradox here. A live bomb and a dud differ so they have a different impact on various other objects. This difference may be observed. You may only prove a paradox if you assume too much, including some of the assumptions of classical physics.
But I decided that the most illuminating explanation what is going on involves the double slit experiment. Richard Feynman liked to say that all the surprising new features of quantum mechanics may be understood if you carefully enough think about the double slit experiment. Nothing else is needed and all the new thought experiments are just repetitive games pushed by those who simply haven't mastered the basics, the double slit experiment.
This point is particularly clear in the case of the bomb tester experiment, I may add. I am probably not the first one who points out that the interferometer may be perfectly replaced with the double slit experiment. But among the people who promote the "quantum mechanics is weird" pseudoscience, the percentage of the people who can actually make illuminating analogies and sketch physically equivalent situations is tiny. They enjoy when things are muddy and confusing. Sorry but I think that even Stillconfused lost his interest and became silent when things became clear. This is not a scientific approach to problems: science is still a sort of problem-solving, not a sort of whining about non-existent problems.
Fine. In the double slit experiment (a diagram is at the top), particles (like photons, particles of light) are emitted by a source. They go through a geometric landscape. Most of them get absorbed by the shields. Some of them may go through the two slits. Behind the two slits, you will see the interference pattern, alternating minima and maxima. If you fine-tune the slits, there are perfect interference minima where the intensity is zero: no particles are landing there. In terms of light, these interference minima remain dark.
Now, the only thing you need to change to turn this double slit experiment to a bomb tester is to cover one of the two slits by a bomb. The bomb explodes when a photon hits it. It is such a simple modification of the double slit experiment that I won't modify any pictures. Many photons will hit the bomb and you get an explosion (about 1/2 of the photons that used to make it through the two slits). However, there will still be photons that make it through the modified slit barrier (effectively through one slit).
Because there is only one open slit after you covered the other one by a bomb, the interference pattern ceases to exist. You get a boring photon distribution at the photographic plate. In particular, the interference minima that used to have a "vanishing intensity" disappear and you get a nonzero intensity there, too. A photon landing to these would-be interference minima proves that there is no interference going on.
But we designed the situation in such a way that the only two options were a "transparent dud" and a "live bomb", interference or not. So no interference means that the bomb is alive. And some photons make it to the interference minima when the bomb is live, thus proving that one of the slits is covered by a live bomb.
The lesson of that "double slit experiment with a bomb" is exactly the same as the lesson of the "bomb tester with an interferometer". The two slits are equivalent to the two arms through which the photons may propagate. The photographic plate with the possible interference pattern is equivalent to the C,D detectors in the interferometer. The photographic plate is a continuum while the C,D detectors are a discrete set of two elements. But it is no tangible difference. The interferometer is built in such a way that the D detector counts "all the photons near the interference minima" while the C detector counts "all the photons near the maxima".
Great. A bomb covering the slit destroys the interference pattern. When it happens, you may get a photon around the minimum. If you do get a photon there, it proves that the slit was covered i.e. the bomb was live. Is it mysterious? Well, it is exactly as mysterious as the normal double slit experiment. The discussion of the double slit experiment with one slit covered is a basic part of the lessons, see e.g. Feynman Lectures on Physics.
"Why" is it possible that we can sometimes find out that the slit is covered without an explosion? It's because the covering of the slit doesn't affect just the small region around the covered slit. It doesn't affect just the question whether the bomb will explode. It also affects the "whole". A slit covered by a bomb turns an interference experiment to a non-interference experiment and that obviously has implications for the photographic plate and the patterns that we see there.
When I emphasize that the covering of the slit doesn't affect just the "explosion of the bomb" but also the experiments done with the "whole", it may sound like some kind of "non-locality". But this fact only looks "non-local" if you think fully classically, if you assume that the observable quantities actually have well-defined values prior to (and between/without) the measurements. In quantum mechanics, they just don't.
The description of the double slit experiment (e.g. quantum electrodynamics) is perfectly local in the quantum sense. For example, in the Heisenberg picture, the operators are evolving according to differential equations (the Heisenberg equations of motion) that guarantee that the operator at a spacetime point may be fully calculated from values of fields operators in the (filled) past light cone of that spacetime point. You don't need to know the values of any field operators at spacelike-separated points! And all influences may be extracted from probabilities of various measurements and those are given by the expectation values of projection operators.
Because the projection operators may be written as functionals of quantum fields in a region and because the quantum fields at the present only depend on the past quantum fields in the past light cone, it follows that all the probabilities also depend on operators (and measurements) that were made in the past light cone, not spacelike-separated, faraway ones! This strictly defined locality perfectly works in quantum electrodynamics. But the underlying theory is a quantum mechanical theory in which the probabilities aren't calculated as the trivial sums as in classical physics. In quantum mechanics, you need to calculate with the probability amplitudes which are the sums and then square their absolute values. There is always the quantum interference. And this quantum interference means that some probabilities are affected by "the arrangement of matter around the bomb-reserved slits" and by something else even though the bomb itself is not detonated. The detonation is about \(|c_2|^2\), the intensity near that bomb-reserved slit, but the probabilities for other observations also include mixed terms like \(c_1^* c_2\) which are nonzero even if the bomb doesn't explode because the explosion of the bomb is governed by the probability \(|c_2|^2\) but \(c_1^*c_2\) is simply a different thing! On the other hand, the influence of mixed terms like \(c_1^* c_2\) on future events doesn't indicate a nonlocal influence between the places 1 and 2 (an action of 1 on 2 or vice versa). Instead, 1 and 2 simultaneously cooperate to influence probabilities in the intersection of their future light cones. Influences are always operating in the timelike directions (i.e. by subluminal speeds) only! Quantum interference is a new, "deeper" kind of cooperation of "possibilities" that exist at two points (or possibilities for general, often faraway, values of general observables) but the chronology between all participating causes and all participating effects is always respected.
In the case of the double slit experiment (and all other experiments in quantum mechanics), you may get more photons at some places (the former interference minima) if you actually cover some slits (e.g. by a bomb) and make the overall picture darker! This is a basic novelty that is impossible in classical physics. In classical physics, a greater number of covered slits (and additional bombs) would mean that fewer photons make it to any point of the photographic plate. But this pointwise inequality is totally refused in quantum mechanics where you get a higher number of photons at the places close enough to the interference minima.
You should think about the experiment with the "slit blocked by a bomb" and you should try to ask all possible questions that you were asking about the bomb tester experiment. And you should answer them. Everything is very clear. The conclusion is that the examples of "weirdness" that some people see in the bomb tester experiments are still the same "surprises" one may find in the double slit experiment. Absolutely nothing new is being discovered here. Just the names and shapes of some components are slightly modified.
So you have this whole industry of anti-quantum zealots that keep on inventing new variations of the very same thing (often some trivial evolution of 1-4 qubits), pretending that they are contributing to the progress in quantum mechanics. In reality, they are still doing exactly the same thing with different words and their confusion proves that they are still confused about the absolute 1925 basics of quantum mechanics, namely about the double slit experiment or the very existence of quantum interference. They are literally playing the quantum counterparts of computer circuit games in which 1-4 new bits are calculated from 1-4 bits of the input. Can you imagine thousands of "would-be revolutionary" papers about calculations with 1-4 ordinary bits? The quantum case really isn't too much harder. Nothing new has ever been found in the subsequent 96 years by these "quantum mechanics is weird" masturbations and all the people who are doing this stuff in order to be promoted in articles about "quantum weirdness" are just defenders protecting fog, obscurantism, and stupidity, idiots who do everything they can to keep mankind (including themselves) mentally stuck in the 17th century.