JesseM said:
The amount of "information" in a given observation is model-dependent, there's no reason at all your model can't involve facts which are totally impossible to measure empirically.
I agree, yet such "facts" are scientifically useless, so let me reframe what I'm saying. Assuming that we agree that "information" in an interesting and useful scientific context is precisely information that is dynamically active, by which I mean has predictive content, then my earlier points go through. That's what I meant by the angels. But my point about the Bohmian trajectories being more of an illusion of information than real information may emerge better after I've established my more important claim, which is that CI and even classical physics are quite capable of navigating the concept of partial which-path information, so I won't go on about the uselessness of the Bohmian trajectories as anything but a philosophically interesting but dynamically sterile ontology.
And since there's no way to measure the microstate of any large real-world system, in practice the microstates are dynamically inert too, even if in principle they are measurable (or so says our model, we have never actually demonstrated this).
No, that is not what "dynamically active" means. There is no need to measure a microstate for it to have dynamically active consequences, its dynamical significance will rest on its measure not its measurability (an unfortunate similarity in words-- by "measure" I mean here something much more akin to "statistical weight", the dynamically active degrees of freedom there).
Are you saying there is some radical difference in our conclusions about "information" depending on whether the true state in question is "impossible to measure in practice, but theoretically measurable in principle according to our model" or "impossible to measure in practice, and also theoretically in possible to measure in principle according to our current model"?
No, I am saying there is no information in the Bohmian trajectories expressly because we calculate no dynamically active or predictive consequences of them, which also means there is no information there, it's just a longwinded label for the information already present in that photon via conventional means. None of this has anything to do with whether or not one can actually measure a Bohmian trajectory, it has to do with the generic nature of such objects, which mean they do not contain any internal degrees of freedom that induce any dynamical activity or changes in any predictions we make. They are just plain not information, and not because we can't measure them. I never said the problem was that we couldn't measure them, so the analogy to unmeasurable microstates misses the mark. Unmeasurable microstates are still dynamically active expressly because they have varying statistical weights that impact our predictions. Bohmian trajectories do not generate any such property, instead they merely echo that property which is already present in the standard formulations. By adding nothing new, they are not information.
What does information as predictive vs. information as "ontological" (in the same sense that
probability can be seen as ontological) have to do with this?
It has everything to do with it. If one wants to use weak measurement to learn something about a system that is physically active and relevant to predictions, then one wants to know if Bohmian trajectories access more of that type of useful information than other approaches do. I am saying they do not, so the only kind of information they would ever access, regardless of how weak the measurement, is the ontological but physically sterile kind. And I'm characterizing that kind of information as trumped-up labeling information. Shakespeare told us why that is not actually information at all, when he said a rose by any other name... I'm changing that to a rose by any other ontological description...
Information theory and game theory can just be viewed mathematically as axiomatic systems, it's only when you get to an interpretation of what the symbols "mean" that the question of predictive vs. ontological could be an issue.
This is the same problem again-- the predictive elements don't care what the symbols mean, they only care about their dynamical involvement (their "statistical weight" in the microstate example). It is ontology that is about what the symbols mean, but when you take what the symbols mean, and project them onto their dynamical significance, all that meaning just projects into a "longer label". That's exactly what I'm saying happens to Bohmian trajectories when you project all that meaning onto their actual physical, dynamical, predictive consequences.
And I don't think you'll find any widespread agreement that your interpretation of the meaning solely in terms of predictions is the only coherent or allowable one (especially since when dealing with statistical mechanics you'd be dealing with 'predictions' that are impossible to test in practice even if they are testable in principle).
This must be viewed as the most central basis of the scientific method, the idea that the logical syntax of scientific epistemology is testing out predictions. Everything else is just labeling: electrons, charges, fields, it's all just placeholders for the unique predictions that are being made about these non-unique ontological entities. We are basically kidding ourselves that what we think of as the meaning of these words has anything to do with science, it's all philosophy. Science is just the syntax that combines these labels into predictions, that's why we can use Newton's laws or Hamilton's principle or D'Alembert statics etc., and it's all the same theory and all the same science. Ontological entities are important because they can help us do the syntax correctly, but only the syntax is the science. Bohmian trajectories are never going to tell you anything about an experiment that isn't just baroque embellishment, it doesn't matter the nature of the observations.
Maybe you agree with this, I just often get the sense that people sometimes think if they just find the right kind of observation to do, suddenly Bohmian trajectories will expose more dynamically active information than the other approaches, getting at some kind of underlying truth whose recognition improves our predictive power. I read that mentality into your question, perhaps it wasn't there.
What is "unitary apparatus"? Do you mean the pure state of apparatus + photons?
Of course, what else is unitary here?
If so why bother with the apparatus, why not just worry about the pure state of the photons themselves?
Because we are never testing any photon pure states, the calculation we are testing very definitely results in a testable mixed-state outcome. This is where the density matrix comes from, the projection of the unitary object here. The guts of what happens is in that mixed state, not in the individual outcomes, those are a kind of necessary evil that we hope the statistical errors of cancel out in a long experiment, and not in the propagating wavefunctions in the apparatus. The calculated mixed state is what is being tested and the experimental mixed state is what is being plotted to perform that test, albeit to within experimental and statistical uncertainties that stem from the technicality that we cannot actually access the experimental mixed state but we get as close as we can by aggregating individual outcomes over many trials.
And what subspace are you projecting on specifically? That was the whole point of my question "what density matrix".
It depends on the question. One subspace of importance is the idler photon, and the basis is which-way information. That density matrix encodes how much which-way information we have about the signal photons, and will be decisive for anticipating the kinds of behavior we will see in the coincidence counts in the x basis.
Why would you need a density matrix for the two-particle system? Are you assuming some classical uncertainty about the pure state of this system?
You are talking about a different state altogether from what I am, this must be the problem. The only time we have a pure state for the two-photon system is before the photons meet the detectors. None of the interest in this experiment focues ontime evolving the pure state of the two photons across the vacuum of the apparatus, because the initial state from the BBO is very complicated and difficult to characterize, and the subsequent propagation is trivial. The state of interest in this apparatus is its final state, that's what is getting plotted and analyzed, and it has a very simple structure-- it is a mixed state of coincidence counts that either either exhibits interference or it doesn't (no further specifics of the x-basis are usually included, being rather difficult apparently), and a state of the idler photon that either exposes or erases which-way information.
So the theoretical state of interest here is indeed a mixed state, not a pure one, and it has the structure of a mixture of |D01>, |D02>, which show interference, and |D03> and |D04>, which do not. These states are states of coincidence count, and have relative probabilities that are all 25% in the standard setup. That mixed state is predicted and observed (the way mixed states are always observed, by building up a statistical aggregate of individually perceived outcomes), and that's the importance of this experiment.
Your suggestion is to change the apparatus to generate a different kind of mixed state, where we cannot automatically associate some of the states with interference patterns and some with not, because we won't have perfect erasure. That just means the states will be something like |D01,I>, the 01 coincidence counts that also show interference, |D01,N>, the 01 coincidence counts that do not show interference, and |D03>, which does not show interference if you leave the D3 path alone. Then there's the same for 02 and 04 of course. Those extra states will be there because of the imperfect erasure, and the mixing fraction of each of those states can be calculated from the apparatus. The outcomes will just have things like 01 coincidence counts over an x basis, but you will be able to check if you have the right combination of interfering and noninterfering contributors just from the shape of the observed distribution over x. This is the part where I said Maxwell's equations are all you need, because the statistical mixed state is a classical limit. Better yet use the observed results from the way the experiment is now done to understand the form of the x-functions, so you don't have to calculate them at all, just combine them with the expected fractions of |D01,I> and |D01,N>. There's no need for any Bohmian path labelings here, you have complete information already.
There is a classical uncertainty there, due to the decoherence of the apparatus. You certainly do
not have a pure two-photon state in this apparatus!
Or as I speculated above, are you thinking that we have a wavefunction which includes the entire apparatus as well as the particles (something that is not ordinarily done in practice when making calculations about experiments like this, obviously) and then talking about the reduced density matrix for the particles alone?
That is what is
always done, in any experiment of this type, except of course you don't spend any time thinking about the entire apparatus wavefunction, you go straight to the mixed state projection onto the two photons via the coincidence count measurements. All of quantum mechanics is calculated and tested on mixed states, pure states are never anything but initial conditions in the quantum mechanics calculations that get tested. Think about that.
Why not? Presumably any good definition of "partial which-path information" would be one that would allow you to calculate it in practice in the usual style, not one that requires you to have detailed information about the quantum state of the entire apparatus, which is impossible in practice.
Nothing I've said relies on any information other than coincidence counts in the x basis, and the which-way/erased eigenstates of the idler photon.
You can't just make up new operators using verbal formulations if you can't give a general definition of what they mean in terms of other known operators, or equivalently in terms of what the wavefunction would be when expressed in that basis after an arbitrary measurement.
Of course you can, all you need is to be able to assert the eigenvalues and eigenvectors, and you have a perfectly good operator. I can do both very easily (the eigenvalues are "which-way" and "erased", and inspection of the idler photon paths gives us when you have which-way information and when you don't, so there's the eigenvectors), so the "which-way" operator is perfectly admissible.
Take my example of the idler being detected at a detector somewhere midway between D1 and D3--for a specific position of this detector, could you calculate what the coefficients would be if the wavefunction is expressed in the which-way basis?
Yes, I already said how. It's purely classical, just use Maxwell's equations. Or even better, observe it rather than calculate it-- put left and right circular polarizers in slits A and B, and measure the polarization at your detector. Voila, if it's elliptical, the linear part is erased and the circular part is which-way.
If you can't calculate such values for specific problems, then talking about a new operator is just verbal kerfuffle with no clear way to translate it into a technical definition.
Fortunately, that is hardly the case here.
Not in your terms it wouldn't, because it would only give you probabilities about hidden variables which are "dynamically inert". Just like someone who believes the religion that says every pin must have 1-1000 angels dancing on it can make a statement about probability, like "there is a .05 chance the number of angels dancing on this pin is in the range 1-50", that have no meaning for someone who doesn't believe that religion.
It sounds like you think that disagrees with me, when I feel it could have been lifted almost verbatim from much of what I've said.
In any case, any definition of "partial which-path information" should just be a function of where the idler is detected and could be used by anyone, including a Copenhagen advocate, even if the CI doesn't see this "information" as quite akin to classical information which is interpreted in terms of classical probabilities (in quantum computing there is already a notion of
quantum information which differs from classical information). The CI wouldn't have to use the word "information" to label this function if you're so hung up on that word, he could call it something totally different, like a "complementarity parameter" that determines how wavelike vs. how particle-like the signal photon behaves when it goes through the double slits. Would that make you happier?
Again, you seem to think this contradicts me, but this is what I'm saying. I said that moving detectors around to get partial information is a fine thing to do, relatively easy to calculate, and frankly I think the results you'll get are pretty obvious but it might make for a nice experiment anyway. What's more, I was saying that I don't think Bohmian trajectories add anything to the issue, they're not going to give us some great new way to access the partial information that isn't pretty apparent in a Copenhagen or even classical approach.
The CI can't meaningfully talk about the "probability" the signal photon went through one slit or another in a case where the idler wasn't at D3 or D4, since it doesn't include any hidden variables for the "true" path a particle took between measurements.
The CI has no difficulty providing language to this situation. Take the example I gave with a polarization measurement on the idler. Let's say you decompose it into 50% left circular and 50% linear, the phase doesn't matter. The CI would then say, even if individual quanta were sent through (it matters not), that 50% went through the left slit, and 50% did not have a meaning to which slit they went through. It's no problem at all.
Or does "no difficulty" mean you think you already have a clear idea of how "partial which-path information" should be defined, so if someone gives you a position X where the idler was detected this definition can be used to compute a quantitative answer for the amount of "partial which-path information"?
Yes, this one.
If the latter I'd like to hear your definition, either in terms of an equation or at least a sufficiently clear description that any good quantum physicist would be able to translate your definition into something mathematical (which I think would be true of my suggestion of how to derive it in terms of fractions of Bohmian paths).
So I've done that now. And I agree that it would have some translation into fractions of Bohmian paths, I'm just saying that the translation is one-to-one, there's no new information or insight in it that's not in CI. I certainly expect that in the example I gave, the Bohmian trajectory result would be 75% left slit and 25% right slit, and again all the information in those numbers would be purely classical, masquerading as quantum information that is not in fact there.