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There is no non-locality and no interaction-free measurement in the bomb tester

Anti-quantum zealots are deluded because they think that a whole history or classical trajectory (in the past) may be extracted from a measurement at one moment

I had some exchange with Petr H. about his claimed "non-locality" in QM and the delayed choice quantum eraser experiment that I have discussed in quite some detail several times (click at the hyperlink to get the most extensive two discussions in 2018).

Recall that the silly conclusion that "there is some action operating backwards in time" in the quantum eraser experiment is just an artifact of a totally irrational psychological (or psychiatric) disorder of some people. When they read the probabilistic distribution for a discrete variable and a continuous one, \(\rho_j(x)\), they always imagine that the discrete variable \(j\) must be the primary switch, the cause, something that decides about the functional dependence of the probability on the continuous variable \(x\) (in particular, whether the probability as a function of \(x\) has one interference pattern, the opposite one, or none).

But that's completely stupid because even in this mixed, discrete-continuous, case, we can equally well read the probability distribution for 2 variables in the opposite way: \(x\) may be decided first while the probabilities for various values of \(x\) are the "inclusive" ones (and indeed, a photon may land on a continuous photographic plate first) and then the relevant probability distribution is reduced to a finite collection of probabilities simply by substituting the right continuous (just measured) value of \(x\) into \(\rho_j(x)\): we get some probabilities \(P_j\).

The primary irrational trait that makes the people say that the Elitzur-Vaidman bomb tester experiment is weird or acausal or measures something non-locally or tests a bomb without touching it etc. is completely equivalent to the psychiatric disorder from the quantum eraser experiment. I have previously pointed out that Vaidman, one of the two fathers of this thought experiment, has admitted that it is misleading to say that an interaction-free measurement is taking place. You can't really measure things without interactions; and you can't affect anything nonlocally. Measurements always need a contact of the apparatus with the measured object (and an influence of the measured object on the apparatus); and all allowed influences in Nature are local (with the possible exception of some debatable types of non-locality in the Hawking radiation, quantum gravity, or string theory but again, even most of those non-localities are illusory artifacts of a description that unnecessarily "creates" some non-locality).

Fine, so let us describe what the bomb tester experiment does once again.

The source A sends a photon to the interferometer at some point, that photon/beam splits to two equally strong beams in the lower left semi-transparent mirror. These two beams have the same intensity and a very particular relative phase. After the beam is split, the photon either goes through the "B arm" (the lower side and then the right side of the rectangle); or the "non-B arm" (the right description of this "or" requires the quantum complex superpositions). The letter B at the bottom is the bomb which may be either an activated bomb with a detector (which absorbs the photon and makes the bomb explode); or a "dud", a transparent model that just looks like a bomb and allows the photons to get through just like in the vacuum.

Now I will already be deviating from the irrational stories about the experiment that do everything they can to confuse others (and the author of these dumb explanations, too). Nature is perfectly local and causal and because the bomb either explodes at the given moment or it doesn't explode (we know the delay between the birth of the photon at A; and the possible explosion, it is given by the distance), it means that right after that moment, we are doing the first measurement. It is a measurement of one bit, a binary observable: the bomb has either exploded or not. There is no non-binary outcome here; I am sure that the transsexual, pansexuals, asexuals, and other non-binary sexuals will be happy about this fact.

You may see that only the first (lower left) semi-transparent mirror is relevant for the prediction of the probability of this first outcome (explosion or not). What the photon does afterwards cannot affect our predictions of the first observed bit (explosion or not). If you think about it, it is totally common sense, if you have been capable of reading some TRF blog posts, you would also be able to make this trivial conclusion, and it's the "quantum mechanics is weird" whackodoodles that want to obscure this basic point (that the arrangement of the other three mirrors and detectors is completely irrelevant for the prediction of the probability of exploded/not)!

Great. Because I said that the first, lower left, semi-transparent mirror splits the beam to two equally strong beams, it means (it is really the same statement) that the probability is 50% that the photon goes to the lower, horizontal arm which contains B (the live bomb or the dud). So with the probability of 50%, the bomb explodes; with the equally high probability of 50%, it doesn't explode. You have made your first measurement.

This looking at the fate of the bomb, whether it exploded or not, was a measurement (of a single bit) done in the vicinity of the bomb-or-dud, whether you like it or not. If the bomb exploded, it proved that the location of the photon was near B because the photon hitting the bomb is a necessary condition for the explosion. In that case, the photon is absorbed and we may successfully predict that nothing will be detected either in the detector C or in the detector D: no photon continues to fly after the explosion if there is an explosion because the explosion requires the absorption of the photon. Fine?

But the measurement in quantum mechanics always modifies the measured object – it does so by changing the wave function. The wave function (describing our knowledge or belief about the whole measured composite system or the whole world) collapses to the eigenstate of the measured observable, the projection on the appropriate "eigenspace". If the bomb has exploded, the post-explosion wave function is trivial and contains no photon moving through the interferometer (the bomb must explode a little bit away from the path, otherwise it would surely send a lot of photons and other stuff into the interferometer).

However, in the remaining 50% of cases when the bomb hasn't exploded, things are tougher. The non-explosion of the bomb doesn't prove that the bomb was a dud (the photon could have missed it instead); and it doesn't prove that the photon has missed the lower side with the B (because the reason of the non-explosion could have been that the bomb was a dud). It's hard but we may keep on using the Schrödinger's picture. The detection of silence (non-explosion) – which is still a measurement, although a seemingly "passive" one – brings the wave function into a superposition of the possible parts. The superposition contains a term with an activated, live bomb (but that must be multiplied by a photon going along the left arm, and therefore avoiding the B arm at the bottom, otherwise there would have been an explosion); and a term in which the bomb is a dud. It is just the amplitude for "the bomb is live and the photon was in the bomb's arm" which is reduced to zero by the measurement of the silence (assuming an orthogonal basis in this sentence).

The post-non-explosion wave function is therefore a superposition of the three allowed terms (but live_bomb-photon_near_B must have a vanishing amplitude; you may also derive it from the orthogonality to the wave function in the case of an explosion). You may find this wave function by taking the perspective of an external experimenter who knows whether the bomb is activated or a dud; and who observes the interior observer, too. When done properly, you confirm the result that out of these 50% non-explosion cases, one-half of them (25% of the whole) ends with a photon in the detector C; one-half of them (25% of the total) ends with a photon in the detector D.

Some time after the non-explosion, we detect another bit of the information because the photon lands either in C or D. As the results summarize it, the outcome C leaves it ambiguous whether the bomb is live or a dud. However, one can prove that if the bomb is live and it doesn't explode, then the photon must land at D, not C. Because we said that an exploded bomb produces neither C nor D, we may also invert this proposition: If we detect a photon at D, we know that the bomb is live (yet unexploded).

Is that an action backwards in time? No. We could reverse the reasoning because the probabilities are zero for live+C and also for dud+D, i.e. they vanish in the whole column and the whole row of the nonzero entry live+D. But just because the probability matrix of the combined results has these vertical and horizontal zeroes doesn't mean that the "detection at D" is the cause that has made the bomb live. Obviously, the bomb may have been assumed to be either live or a dud well before the possible detection at C or D. After all, I described the measurements chronologically.

Is that an interaction-free measurement of the livelihood or duddiness of the bomb? Again, no. There are no interaction-free measurements in Nature, it is a contradiction. If you used this interferometer setup to check whether your bombs are live, many of them would explode, it is the 50% option (the simpler outcome of the first binary measurement, explosion-or-silence) that we started with. If you were willing to sacrifice 50% of the bombs and if you only wanted to make a statement about an interaction-free measurement of a single bomb which was lucky not to explode, it is still wrong to say that the livelihood of the bomb was measured in an interaction-free way.

As I have already indicated, whether the bomb has exploded or not was determined from the detectors of explosions placed around the bomb, and the result "silence at the dangerous time" was just one possible result of the measurement of some observables constructed out of quantum fields in the B region (around the bomb).

But indeed, you may object: We don't really need to know whether the bomb has exploded. It is enough for the photon to land in the detector D and then we can say that the bomb is live! So we only make the measurement in the D region and when we get some photons there, it proves that there was a live bomb at B and the photon's trajectory avoided B. The first assertion is legitimate, the detection of a photon near D does prove that B was live (and the distance between B and D doesn't make it nonlocal, the events at the two places are correlated with one another because a photon could have moved through both); but the last assertion, the photon avoided the B arm, is deceitful. Why? Because it is trying to erase a part of the wave function retroactively, to rewrite the history. But in the rational usage of quantum mechanics, you should never rewrite the history.

As we said, the photon created at A left the first, lower left, semi-transparent mirror with equal amplitudes in the two arms. At that moment, and before the explosion-or-silence moment, it was just manifestly wrong to say that the photon was forbidden in the lower B arm. After we heard the silence (non-explosion), assuming that it was one of our observations, we reduced the wave function. The result "silence, not an explosion" was decided randomly, a new bit of information was created, and the "live-B" option was eliminated. But the other three basis vectors – and all nontrivial superpositions different from a multiple of "live-B" – were still allowed. The observation of the silence has modified the wave function, including the probabilities of the B-or-nonB paths.

If we decide not to observe the explosion-or-silence bit, we may talk about the wave function for the photon's location before the C-or-D detection. The interpretation of the D detection says that "before the photon got to the last mirror, it had to be on the non-B path". But that's not what the D detection actually says. The D detection corresponds to a superposition of the photon "on both paths" with the same probabilities and a D-specific relative phase.

The problem is the same as in many other stupid "paradoxes about QM" which are not real paradoxes. The people assume or conclude that the "photon went to D i.e. was in a supersition B plus non-B" and also "it went through the non-B path". These two states are non-orthogonal but distinct states in the Hilbert space. The non-orthogonal property means that they are not mutually exclusive. But they are still different vectors in the Hilbert space which means that you can't assume two such initial states to be the right starting point for further predictions at the same moment. The predictions resulting from different initial states generally differ.

I mentioned that it is wrong to retroactively rewrite the history. Let me describe the same idea in another way that might be clearer. Is it right to say that the detection of a photon at D proves that the photon went through the non-B arm? No because the measurements can't be used to reconstruct the whole history. The point is that observables at different moments generally refuse to commute with each other so you can't measure all of them through a measurement at one moment. In particular, if you wanted to say that "the photon went through the non-B arm", it would be equivalent to the statement about the location of the photon in the past. But what we actually measured by the detection at D was the location of the photon at the very end! And because of the semi-transparent mirror between C and D, the operator for the location of the photon "when it was in B or non-B arm" doesn't commute with the operator of the location of the photon "whether it ends in C or D". The corresponding bases are linear superpositions of each other, the usual kind of a 45-degree rotation. The commutator is "maximal" for a 2-state system; the situation is isomorphic to the nonzero commutator of \(j_x\) and \(j_y\) for the electron's spin.

It's really just a classical yet completely impractical retroactive interpretation of the measurements that creates all the weirdness. The point is that when we measure something, we always measure it in order to know something about the present and the future. The purpose of any measurement is to fix the predictions or expectations about all the future events (as described by future measurements!). We shouldn't really expect the measurements to produce an increasingly fine description of the history. That is really impossible in quantum mechanics because the operators for the "same observables" but at different times generally have nonzero commutators!

That doesn't mean that science must sacrifice the research into the question whether dinosaurs ever walked on Earth before they went extinct. But the reconstruction of the answers to the question "whether dinosaurs have ever lived here" (the answer is usually extracted from a recent measurement of the fossils) is only possible because we are extracting values of observables that basically behave classically and their commutators are effectively zero. At some level of precision, the history of species on Earth looks like a classical history in which all the commutators are negligible. But that approximation is maximally wrong in the "bomb tester" experiment because the location operator "B arm or non-B arm" and the location "C or D, the final detector chosen by the photon" maximally refuse to commute, the bases are rotated by 45 degrees. So that is where the reconstruction of a whole history of locations from a measurement at one (late) moment is impossible. You just can't measure the locations at the two moments simultaneously much like you can't measure \(X\) and \(P\) (or \(j_x\) and \(j_y\), more precisely) at the same moment! Instead, any such a measurement of location should be interpreted as the measurement of the "latest location" only which tells us nothing about the locations in the past.

Any statement of the type "the photon went through a particular arm AND landed in the D detector" is wrong simply because it is in a full-blown conflict with the uncertainty principle (if no measurement took place in between the two relevant moments). The two values of the locations at the two moments can't simultaneously have sharp eigenvalues because the operators don't commute!

Again, like all other experiments that are possible in Nature, the bomb tester experiment is 100% compatible with locality, 100% compatible with causality (the chronological order of the cause and effect), 100% compatible with the statement that you won't learn anything about the "region of the bomb" if your apparatuses and photons never touch that place, and any counterintuitive results follow from the incorrect interpretation of "late measurements" as something that clarifies a whole history. Late measurements can't do such things and the precise trajectory of the photon cannot be said to be known because the locations at different moments refuse to commute with each other, just like \(j_x\) and \(j_y\) do. The measurement of the location of the photon only tells us something about the most recent location and renders all the ideas about the previous locations obsolete and overwritten. Quantum mechanics is not weird – it is just many people who are dumb.

And that is the memo.

P.S. 1: In other words, after you detect it at D, assuming that you didn't observe the non-explosion, you can't say that the photon strictly avoided the B arm. Instead, the arm the photon has chosen remains an ambiguous history, the photon was in some unknown superposition of both arms which is how it was created. If the non-explosion wasn't a measurement, you can however predict that later measurements of the explosion-or-not would say "not". But... Only measurements have the right to "collapse" the wave function to a particular classical answer and the measurements of locations (e.g. the C/D detection) only tell us about the latest location, not the previous ones which are overwritten. The sharp statements like "location was strictly this" are only possible right after the measurement, and for the near future when the predicted location remains 100% certain. You shouldn't extrapolate the sharp statements about locations or other observables into the past from a measurement. If the B's non-explosion is counted as the first measurement, then some reduction happens at the moment of the non-explosion. But that collapse is not quite enough to rule out the B arm. When both the non-explosion and D-detection are counted as measurements, their combination may indeed be used to determine that the location of the photon was in the B-arm in the middle. But we can't say that we didn't do a measurement in the region B in that case! I want to emphasize that we don't need to add anything new to QM to deal with the bomb testers. The usual rules of QM are being violated those who invent "weirdness", namely 1) the uncertainty principle - you can't measure two non-commuting observables simultaneously, 2) it always matters whether some event is a measurement because measurements/observations are the only events that discontinuously modify a wave function. These collapsed wave functions evolve by Schrödinger's equation up to the next measurement.

P.S. 2: Stillconfused has correctly argued that I haven't really discussed the surprising point that "one can identify a live bomb without an explosion" even though the explosion happens exactly iff the photon is absorbed. I wrote the following explanation that visualizes the interference as a double slit experiment:

Hi, I understand your surprise.

The two arms of the interferometers are like two slits, OK? And the photon is creating an interference pattern. With a dud, D is the interference minimum of that interference pattern where the probability of landing is exactly zero, OK? C is the interference maximum.

But the interference pattern disappears and you one gets nonzero probabilities both in C and D if the real bomb is inserted (it's effectively as if the "slit", the arm with B, is closed) which is why a nonzero detection at D proves that the bomb is live - the interference patern was destroyed and the intensity-zero interference minimum is no longer interference zero minimum (detector D).

Does it make sense?

Classically, you wouldn't distinguish C and D because the spacing of the interference pattern is classically infinitely short. So really classically, you would have detections both in C and D at all times. However, quantum mechanically, the probability of detection at D becomes zero with a dud, a perfect interference.

So in this sense, the surprising novelty of QM is that it is capable of producing the exact zero at D for a dud. THIS is the surprising point of QM that isn't possible to reproduce in a truly classical theory. Unsurprisingly, every detail in the arrangement modifies or destroys the interference pattern, so the perfect-interference prediction of "zero intensity at D" breaks down with a live bomb.

My real point is that, as Feynman liked to say, all the surprising things in QM are contained in the double slit experiment. Your surprise is equivalent to the surprise that the double slit experiment has minima with both slits open, but has no minima when one slit is closed, and even when one slit is closed, you can get photons that aren't absorbed by the closed slit but that do land at the place that would be the perfect interference minimum if both slits were open.

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