Exploring Photon Trajectories in Double Slit Experiments

[SpectraCat]

Oh yeah! Ever heard of the LHC and electromagnetic acceleratation??

No no no, of course you are right... I don’t know what I was thinking on... too fast, too enthusiastic... ()

There’s always 'something' with this darned experiment, isn’t it??

Without knowing anything about it, I can almost guarantee you that it will be impossible to time photons thru the slits – without disturbing them to 'non-interference pattern'. And I’m pretty sure it’s the other way around with electrons, no problem with timing – but then you don’t have any support from Einstein’s c ...

This is nuts!
I can’t say for sure, but shouldn’t you be able to 'surpass' HUP by scaling up the whole experiment to a size where this doesn’t matter – ie if your resolution is 1 second/meter, you drive 1000 meters and that 'resolution' will be sufficient. I think...

But as you pointed out - it won’t work with electrons anyway...

Many thanks for the help!

I am not so sure the timing is the issue. I wonder if the following might actually work. Take a metal-double slit and super-cool it. Then measure the inductance-current in the *entire* double slit apparatus from the electrons (ions would work too), passing through it, and define the peak of the inductance current as t=0. In principle, there should be no "which-path" information from such a measurement.

I wonder what would be observed at the detection screen? My guess is that you would still see interference, even though you have nominal timing information. This is because the induction will couple the momentum of the electrons in the experiment to those in the conduction band of the metal screen, so although you will have timing information, you won't be able to use that to determine which slit the electron went through.

On the other hand, if you are clever, it might be possible to get which path information from the current measurements. If you connect wires to either side of the super cooled double slit, and measure the current at both ends, there should be a timing difference depending on which slit the electron passes through .. i.e. the response pulse should arrive slightly earlier at whichever side corresponds to the slit the electron passed through. I know this timing-based detection works in principle .. it is used to image electron distributions in photo-ionization and photo-detachment experiments. What I don't know is whether the temporal resolution is sufficient to discriminate between the closely spaced slits in the double slit experiment.

Anyway, for the purposes of a gedanken experiment ... I think it *is* possible to discriminate the timing. Can anyone figure out why measuring the current in the timing-sensitive method I mentioned above would destroy the interference pattern?

 

[Ken G]

Experimentalists do.

The issue here is not the experiment itself, but the way the data was packaged and sold. I never had any issue with doing weak measurements, which is the experimentalist part, my issue was with the "average trajectory" concept, which is not an experimental issue, it is an information manipulation issue. I wouldn't even call it "data reduction", it's more like "data obfuscation."

I agree. But my point is the following: It does not significantly contribute to a demystification of QM, as long as it cannot be generalized to the case of entangled particles.

Yes, we certainly agree that this experiment does not significantly contribute to demystification of QM. You are criticizing what they did not do, and I'm criticizing what they actually did, and the claims they made about what they actually did.

One reason why weak measurements and Bohmian mechanics are more cool than your classical-wave picture is the fact that they can be applied to entangled particles as well.

Maybe yes, but that has nothing to do with this experiment. My point is that, in this experiment, a figure was produced that does not clearly contain any non-classical information, and that's the reason there is no quantum demystification here. Had they done something involving entanglement, perhaps there'd be some interesting Bohmian consequences, perhaps not-- we'd have to see what can actually be done there.

But if it works in the absence of entanglement only, then the significance of it is rather low.

If its output is the same as a purely classical calculation, then its significance is zero. Yet the authors claimed it was something different from the way they were taught quantum mechanics. I'm saying I have not seen a shred of evidence for any of those claims, regardless of whether entanglement issues would make it more interesting.

 

[DevilsAvocado]

I am not so sure the timing is the issue. I wonder if the following might actually work. Take a metal-double slit and super-cool it. Then measure the inductance-current in the *entire* double slit apparatus from the electrons (ions would work too), passing through it, and define the peak of the inductance current as t=0. In principle, there should be no "which-path" information from such a measurement.

Maybe it’s me misunderstanding, but when it comes to getting exact timing for electrons leaving the double slit, I’m 99% it is already doable by utilizing the http://en.wikipedia.org/wiki/Aharonov-Bohm_effect" .

There’s a Time-resolved detection of single-electron interference using an Aharonov-Bohm-ring in this paper http://arxiv.org/abs/0807.1881

https://www.youtube.com/watch?v=sT6OyzJ8Oqw
The movie shows in real-time the gradual build-up of an interference pattern as we count electrons passing through an Aharonov-Bohm-ring. Since only one electron can pass through the ring at a time, the experiment shows that each electron is interfering with itself.

I wonder what would be observed at the detection screen? My guess is that you would still see interference, even though you have nominal timing information.

My guess is that you will see a perfect interference pattern – that could even be 'adjusted' by the Aharonov–Bohm effect!

https://www.youtube.com/watch?v=OgDPK5MLVnE

On the other hand, if you are clever, it might be possible to get which path information from the current measurements. If you connect wires to either side of the super cooled double slit, and measure the current at both ends, there should be a timing difference depending on which slit the electron passes through .. i.e. the response pulse should arrive slightly earlier at whichever side corresponds to the slit the electron passed through. I know this timing-based detection works in principle .. it is used to image electron distributions in photo-ionization and photo-detachment experiments. What I don't know is whether the temporal resolution is sufficient to discriminate between the closely spaced slits in the double slit experiment.

Anyway, for the purposes of a gedanken experiment ... I think it *is* possible to discriminate the timing. Can anyone figure out why measuring the current in the timing-sensitive method I mentioned above would destroy the interference pattern?

As a layman, the only thing I can think of is the detection causing a current at the slits – if this is a 'regular' measurement, it will cause decoherence...

If not – it’s really interesting!

 

[Ken G]

What I don't know is whether the temporal resolution is sufficient to discriminate between the closely spaced slits in the double slit experiment.

The timing difference is of order the slit separation divided by c, call it d/c. To get interference, you need the wavelength, call it l, to be of order d or larger. Since the period of the wave is l/c, this means the period must be of order or greater than the timing you are trying to resolve. You cannot resolve the timing of a coherent wave at scales of its own period. Put differently, your setup is having a little fight between the need for the wave to have a sharp frequency, so you get the interference pattern (that's why lasers are often used), and the need to have timing resolution of order the wave period. You can't get the latter with a sharp frequency, that is a basic wave property.

 

[DevilsAvocado]

Ken G, sorry for asking this really dumb question, but I can’t get it out of my head: Electrons have mass, they don’t travel at c (unless you 'scare' them with a LHC). According to CI electrons doesn’t 'exist' until you measure them, and before that we use the wavefunction to calculate the probability for the electron to 'popup' in a specific location.

Question – What speed does the wavefunction for electrons have (according to CI)?

(no entangled stuff)

 

[SpectraCat]

The timing difference is of order the slit separation divided by c, call it d/c. To get interference, you need the wavelength, call it l, to be of order d or larger. Since the period of the wave is l/c, this means the period must be of order or greater than the timing you are trying to resolve. You cannot resolve the timing of a coherent wave at scales of its own period. Put differently, your setup is having a little fight between the need for the wave to have a sharp frequency, so you get the interference pattern (that's why lasers are often used), and the need to have timing resolution of order the wave period. You can't get the latter with a sharp frequency, that is a basic wave property.

Well, aside from the fact that electrons don't propagate at c (as DA already mentioned), I think you have hit on an important issue. What you are saying is that, if the double slit is of the appropriate size to cause diffraction, then the question of which slit the electron passes through is meaningless. Since the wavelength of the electron is on the same order as the width and separation of the slits, the spatial resolution is not sufficient to determine through which slit the electron passed.

Of course, that analysis assumes that the electron has a uniquely wavelike interaction with the double slit. I see no reason to assume that the other aspects of the experiment I suggested (i.e. supercooling the double slit and measuring the induced current from either side) would do anything to change that. So it seems like there would be no way to get which path info from this gedanken, and thus a normal interference should be observed.

However, I would like to ask the Bohmian's about this experiment. I guess in the Bohmian picture, the pilot wave will diffract off both slits, but each single electron will take a well-defined trajectory through one slit to the screen. Is that correct? Also, in the Bohmian picture, are properties like charge associated with both the pilot wave and the particle, or just one or the other? It seems like charge would be a particle-only property, since charge is usually (always?) associated with localized detection events, but perhaps that is just a poor assumption on my part.

 

[Ken G]

Question – What speed does the wavefunction for electrons have (according to CI)?

This is called the "group velocity" of the wave, it's essentially the speed that the "wave packet" moves if we watch the solution of the wavefunction evolve in time. By the correspondence principle, it has to work fine in classical physics.

 

[Ken G]

Well, aside from the fact that electrons don't propagate at c (as DA already mentioned), I think you have hit on an important issue.

Yes, I answered it for photons, but the electron wavefunction is not qualitatively any different. Wave mechanics is wave mechanics.

What you are saying is that, if the double slit is of the appropriate size to cause diffraction, then the question of which slit the electron passes through is meaningless. Since the wavelength of the electron is on the same order as the width and separation of the slits, the spatial resolution is not sufficient to determine through which slit the electron passed.

Actually, you can tell which slit the electron passed through, but not without disrupting its wavefunction. You generally want to tell which slit it went through while just being a "fly on the wall", not disrupting anything, but this is just what you cannot do. Even quantum erasure experiments involve a change in the wavefunction, during parametric down-conversion.

In the above, I was talking about the period of the undisturbed wave-- you can always time it with arbitrary precision by blasting it with a bright radiation field, but the energy needed will disrupt the coherence of the wavefunction and alter the results.

So it seems like there would be no way to get which path info from this gedanken, and thus a normal interference should be observed.

That's harder to tell. My point was that if you can do the timing, then you cannot get an interference pattern, but I can't tell if your apparatus will allow the timing or not, because I can't tell how much it will disrupt the coherence of the wavefunction. If I had to guess, I'd agree with what you're saying here-- you should still get the interference pattern.

I guess in the Bohmian picture, the pilot wave will diffract off both slits, but each single electron will take a well-defined trajectory through one slit to the screen. Is that correct?

Yes. I can tell you that even without being a true believer in the Bohm interpretation!

 

[DevilsAvocado]

This is called the "group velocity" of the wave, it's essentially the speed that the "wave packet" moves if we watch the solution of the wavefunction evolve in time. By the correspondence principle, it has to work fine in classical physics.

Yes, I answered it for photons, but the electron wavefunction is not qualitatively any different. Wave mechanics is wave mechanics.

Thanks for the answer.

I also looked it up myself, the wavefunction for electrons and photons differ (mass/massless) due to the time-evolution of the field operator on Hilbert space, which is governed by the Hamiltonian.

I.e.:
photon wavefunction = c
electron wavefunction < c (and the actual speed depends on the energy)

More info in this thread: https://www.physicsforums.com/showthread.php?t=152053"

 

[Ken G]

Yes, the dependence of speed in a vacuum on energy for a particle with mass is a lot like propagation through a dispersive medium. This is why the connection between frequency and wavelength (which is how you find the group velocity) is called a "dispersion relation." In effect, a massless particle propagates dispersionlessly through vacuum at c, but a particle with mass propagates through vacuum as if it was slowed by interactions between its mass and the vacuum (I'm not formalizing this into a theory, just describing a picture). The result is essentially a dispersion relation, which says (nonrelativistically) that the connection between energy and momentum (frequency and inverse-wavelength) is E = p²/2m (which is a lot like saying frequency scales with inverse-wavelength squared divided by m). The group velocity is given by taking the derivative of the frequency with respect to the inverse-wavelength, so here that velocity comes out p/m.

 

[yoron]

I agree with Ken. I find all 'light paths' diffuse myself :)
To me a 'photon' has a recoil and a annihilation. We assume a propagation based on the way we've seen the universe macroscopically for a very long time. The difference between a football coming at you and a 'photon' becomes even weirder if you consider that all information you get of that footballs path is by those same photons :)

To me it's simpler to describe in terms of what we can measure, than to build theories pressing forward ideas that doesn't have the same clarity they once seemed to have. Einstein and Lorentz both questions our definitions of motion macroscopically, and QM does it 'microscopically'.

 

[SpectraCat]

Yes, I answered it for photons, but the electron wavefunction is not qualitatively any different. Wave mechanics is wave mechanics.

Yes and no .. photons cannot be described by wavefunctions .. I'd say that is a qualitative difference.

Actually, you can tell which slit the electron passed through, but not without disrupting its wavefunction. You generally want to tell which slit it went through while just being a "fly on the wall", not disrupting anything, but this is just what you cannot do.

Of course .. that was the whole motivation for my gedanken in the first place .. I was trying to understand why the "time-lag" detection of the induced current would lead to localization of the wavefunction on one of the slits.

Even quantum erasure experiments involve a change in the wavefunction, during parametric down-conversion.

I am not sure what you are talking about there ... the only PDC in the quantum eraser experiments I am familiar with is when entangled photon pairs are created at the source. The "erasure" step typically involves a normal optical element, as far as I am aware. In the Walborn setup it's a polarizer, while in the DCQE of Kim and Scully, it was just a beamsplitter. Furthermore, in those experiments the "erasure" is only evident in the two-photon coincidence measurements ... the interference pattern is never restored in the single-photon measurements. This can be understood through an analysis of the phase-relationship between the entangled photons, as has been discussed in detail recently on a few parallel threads here on the DCQE:
https://www.physicsforums.com/showthread.php?t=503667
https://www.physicsforums.com/showthread.php?t=505115

In the above, I was talking about the period of the undisturbed wave-- you can always time it with arbitrary precision by blasting it with a bright radiation field, but the energy needed will disrupt the coherence of the wavefunction and alter the results.

Not sure how that would work for free-electrons ... what I mean is that, while I can certainly see how the light field would perturb the electrons, it is hard to see how individual electrons would change the light field in a detectable way. I suppose in the gedanken-limit of perfect detectors, you could see the small change in the light field, I am just not sure about the details of how that would work.

That's harder to tell. My point was that if you can do the timing, then you cannot get an interference pattern, but I can't tell if your apparatus will allow the timing or not, because I can't tell how much it will disrupt the coherence of the wavefunction. If I had to guess, I'd agree with what you're saying here-- you should still get the interference pattern.

That was the entire point of my question .. the difference between the "single-wire" detection of the induced current, where you get only longitudinal timing to set the t=0 for the travel time to the screen, and the "double-wire lag-time" detection, where you also try to get information about the transverse components of the electron momentum, seems very small to me. That's why I think your comments about the transverse wavelength of the electron being on the same size scale as the slits is the key point for understanding why the proposed "lag-timing" can not work. In any case, I certainly agree that if it *did* work, then there would be no interference pattern. I just can't figure out any reason why the apparently minor change in the detection electronics would affect the wavefunction so drastically. That's why ...

Ok .. I am going to stop arguing in circles now .. I am getting dizzy.

 

[Ken G]

Yes and no .. photons cannot be described by wavefunctions .. I'd say that is a qualitative difference.

Photons get described by wavefunctions all the time. Look at any quantum mechanical scattering calculation, or even the derivation of the cross section of an atomic transition. There are some technical matters given that these are typically done without using any relativistic physics at all, yet they serve quite well to determine these cross sections. Quantum mechanics, and quantum electrodynamics, can be conceived at many different levels, some so abstract that there's never any such thing as an evolving wave function (even the Heisenberg formulation does away with wavefunctions as anything but an index on the different possible evolutions for the operators). I don't know what can be said in general about wavefunctions, but I don't know of any fundamentally different behaviors of photons or electrons in two-slit experiments, unless you explicitly try to take advantage of the electron mass or charge.

I am not sure what you are talking about there ... the only PDC in the quantum eraser experiments I am familiar with is when entangled photon pairs are created at the source.

Yes, that's when their wavefunctions are altered. They cannot be treated as two identical input wavefunctions, so the PDC leaves some kind of subtle imprint on them which is quite important for what ensues.

That was the entire point of my question .. the difference between the "single-wire" detection of the induced current, where you get only longitudinal timing to set the t=0 for the travel time to the screen, and the "double-wire lag-time" detection, where you also try to get information about the transverse components of the electron momentum, seems very small to me. That's why I think your comments about the transverse wavelength of the electron being on the same size scale as the slits is the key point for understanding why the proposed "lag-timing" can not work. In any case, I certainly agree that if it *did* work, then there would be no interference pattern.

Yeah, that's the key isue for the Bohmian consequences. I wish I knew more about the detailed consequences of the apparatus you are describing, because "the devil is in the details" with this kind of thing. Every time we think we have found a way to shortcut the rules, reality inserts some subtle element we didn't anticipate to make sure that we can't.

 

[SpectraCat]

Yes, that's when their wavefunctions are altered. They cannot be treated as two identical input wavefunctions, so the PDC leaves some kind of subtle imprint on them which is quite important for what ensues.

That still doesn't seem to make much sense ... there is no requirement that the photons be identical for the quantum eraser experiment ... they could even have different wavelengths and it would still work just fine. The "subtle imprint" that I guess you are referring to would be the polarization entanglement. That guarantees that the photons maintain a well-defined phase relationship, which allows the reconstruction of the interference pattern based on polarization-filtered coincidence counting. In any case, the situation is very different than for the double-slit experiments we have been discussing here, because there is never an interference pattern observed for the photons that pass through the double-slit.

 

[Ken G]

That still doesn't seem to make much sense ... there is no requirement that the photons be identical for the quantum eraser experiment ...

Hmm, I say the photons aren't identical and you say that doesn't make sense, the photons aren't identical. I can't follow that logic. All I'm saying is that the down-conversion leaves an imprint on the wavefunction which is very important to what ensues, pursuant to the overall issue that you can't get which-way information without messing up the interference pattern, even in Bohmian mechanics unless there's some technicality that has not yet been tested.

The "subtle imprint" that I guess you are referring to would be the polarization entanglement. That guarantees that the photons maintain a well-defined phase relationship, which allows the reconstruction of the interference pattern based on polarization-filtered coincidence counting.

It's more subtle than that. Something causes the signal photons to divide into two sets, which each produce an interference pattern that is slightly shifted from the other, such that when you add them up, you get no interference pattern.

In any case, the situation is very different than for the double-slit experiments we have been discussing here, because there is never an interference pattern observed for the photons that pass through the double-slit.

I completely agree, that's why I've said so little about entanglement in this thread. Others seem to want to keep bringing it up, but I agree it's a red herring in regard to Bohmian trajectories in a simple two-slit experiment.

 

[JesseM]

Commenting on SpectraCat's original post, isn't there still a meaningful sense in which we can talk about "which-path information" in the Bohmian interpretation of a quantum eraser or delayed choice quantum eraser experiment? While it's true that if you look at the position the "signal" photon was detected you can always tell which slit it went through just by looking at the position on the screen, what if the only information you have is the position the idler was detected (and the experimental setup it passed through to get to the detector, like whether there was a polarizer in front of the detector in the basic quantum eraser experimenter), in some cases this information alone would be enough to tell you which slit the signal photon went through while in others not, right? In the delayed choice quantum eraser paper, if you look at the D0/D1 coincidence pattern in Fig. 3 on p. 4, in the Bohmian interpretation would it true here (as with the normal double-slit interference pattern) that half the signal photons making this pattern went through the left slit while half went through the right? And for the D0/D3 coincidence pattern in Fig. 5, in the Bohmian interpretation would it be true that all the signal photons making this pattern went through the same slit? If so, that would be a sense in which detecting the idler at D1 means the which-path information has been entirely "erased" while detecting the idler at D3 means the which-path information has been entirely preserved.

One thing I've wondered about this type of experiment is whether it would be possible to arrange the idler-detectors in different ways so that one could have have "partial" which-path info in some sense that could be quantified, and whether a detector which gave partial which-path information would then show a partial interference pattern (somewhere between fig. 3 and fig. 5), with the closeness to a perfect interference pattern depending on the "amount" of which-path information. Since "amounts" of information are usually defined in terms of probabilities of a given outcome via the information-theory concept of entropy, in most interpretations of QM it would be a bit problematic to talk of partial information since we can't really talk about the probability that the particle "actually" went through one slit or another, but in Bohmian mechanics you can, so maybe this would be a convenient way to quantify partial which-path information and see if it corresponded to partial interference.

 

[Ken G]

Commenting on SpectraCat's original post, isn't there still a meaningful sense in which we can talk about "which-path information" in the Bohmian interpretation of a quantum eraser or delayed choice quantum eraser experiment?

I would say the Bohmian approach always dictates which way any given photon goes, but it can't count as "information" if we cannot extract it. Are you defining "information" by what "the universe knows about itself", or by what we can know about it? I suspect the latter, as that is the usual meaning.

In the delayed choice quantum eraser paper, if you look at the D0/D1 coincidence pattern in Fig. 3 on p. 4, in the Bohmian interpretation would it true here (as with the normal double-slit interference pattern) that half the signal photons making this pattern went through the left slit while half went through the right?

In my understanding, yes.

And for the D0/D3 coincidence pattern in Fig. 5, in the Bohmian interpretation would it be true that all the signal photons making this pattern went through the same slit?

I think so, yes.

If so, that would be a sense in which detecting the idler at D1 means the which-path information has been entirely "erased" while detecting the idler at D3 means the which-path information has been entirely preserved.

I'm not sure what you are saying here, it seems to me that this statement is true in all the interpretations, it's just a fact of the apparatus.

One thing I've wondered about this type of experiment is whether it would be possible to arrange the idler-detectors in different ways so that one could have have "partial" which-path info in some sense that could be quantified, and whether a detector which gave partial which-path information would then show a partial interference pattern (somewhere between fig. 3 and fig. 5), with the closeness to a perfect interference pattern depending on the "amount" of which-path information.

I would expect so, yes. This is because "partial" information would have to look like a probability that you know and a probability that you don't, so if you like, a mixture of a mixed state and a superposition state. But that's just a mixed state of a non-orthogonal basis, so a density matrix with nondiagonal elements that can't be diagonalized, I believe.

Since "amounts" of information are usually defined in terms of probabilities of a given outcome via the information-theory concept of entropy, in most interpretations of QM it would be a bit problematic to talk of partial information since we can't really talk about the probability that the particle "actually" went through one slit or another, but in Bohmian mechanics you can, so maybe this would be a convenient way to quantify partial which-path information and see if it corresponded to partial interference.

I actually think regular quantum mechanics can handle the partial information concept, we'd just talk about the probability that it went through slit 1, the probability it went through slit 2, and the probability that we don't know what slit it went through. In a sense we already have that with the delayed choice experiment if we set up a D4 detector symmetrically to D3 (why isn't it in there?). Then we have a 25% chance of detection of the idler at each of D1-4, so that means we have a 50% chance of not knowing which slit the signal photon went through, a 25% chance we'll know it was slit A, and a 25% chance we'll know it was slit B. I think we have those probabilities in any interpretation.

 

[JesseM]

I would say the Bohmian approach always dictates which way any given photon goes, but it can't count as "information" if we cannot extract it. Are you defining "information" by what "the universe knows about itself", or by what we can know about it? I suspect the latter, as that is the usual meaning.

This seems like a false dichotomy, "information" normally involves the question of how the facts you do know narrow down the range of possibilities for the facts you don't know, even if it is practically or theoretically impossible to determine the single correct truth about the facts you don't know. Consider statistical mechanics, where the thermodynamic entropy of a given "macrostate" (defined by the value of thermodynamic variables like temperature and pressure) is proportional to the logarithm of the number of possible "microstates" (precise specification of the state of all the particles that make up the system) consistent with that macrostate. So, the facts you do know (macrostate) allow you to narrow down the range of possible values for the facts you don't know (microstate), and since there are fewer possible microstates associated with macrostates that have a lower value of (thermodynamic) entropy, we can say that you gain more "information" about the system's microstate if you find it in a low-entropy macrostate than if you find it in a high-entropy macrostate, even though it may be impossible in practice to ever "extract" the true fact about the precise microstate. So I'm suggesting something similar for "which path information" in a Bohmian interpretation, where what you do know (the detection of the idler at some detector) gives you information about something you don't know, but which is assumed to have an objective value (which slit the signal photon went through), and where you can calculate theoretically the set of different trajectories which are consistent with the fact you do know about the idler's detection.

If so, that would be a sense in which detecting the idler at D1 means the which-path information has been entirely "erased" while detecting the idler at D3 means the which-path information has been entirely preserved.

I'm not sure what you are saying here, it seems to me that this statement is true in all the interpretations, it's just a fact of the apparatus.

The reason I discussed this is that SpectraCat had trouble understanding how we could still talk about the which-path information being "erased" under the Bohmian interpretation, since in the Bohmian interpretation even when you get an interference pattern on the screen you can still tell which slit the signal photon went through based on whether it's on the upper or lower half of the screen. My point was that if you throw out the knowledge of where on the screen the signal photon was detected, and only have knowledge of where the idler was detected, then some possible idler detections will correspond to the which-path information being "erased" because they alone tell you nothing about which slit the signal photon went through.

I would expect so, yes. This is because "partial" information would have to look like a probability that you know and a probability that you don't, so if you like, a mixture of a mixed state and a superposition state.

"Mixture of a mixed state and a superposition state" doesn't seem to make sense, wouldn't that just be another mixed state? Are you just thinking out loud or have you seen such a "mixture" defined in a mathematical sense somewhere? In any case I didn't mean anything like that by "partial" information, I just meant putting the detector at some position midway between a position where you completely lose the which-path information (the position of D1 on the last page of the DCQE paper for example) and a position where you completely preserve the which-path information (the position of D3 for example). In that case there might be a way to quantify a "partial" which-path information, and the amount of partial which-path information might correspond to the degree to which the coincidence count resembled the D0/D1 coincidence count (fig. 3) vs. the D0/D3 coincidence count (fig. 5). My suggestion was that the Bohmian interpretation might give a nice way to quantify "partial information" since for every idler detected there will be a definite truth about which path the signal photon took, so you can calculate theoretically the proportion of photons in the coincidence count that went through one slit vs. the other (then the idea would be that the closer they are to evenly distributed between slits the closer you are to zero which-path information, and the closer they are to all having come through one of the two slits the closer you are to 1 bit of which-path information)

I actually think regular quantum mechanics can handle the partial information concept, we'd just talk about the probability that it went through slit 1, the probability it went through slit 2, and the probability that we don't know what slit it went through.

How can regular quantum mechanics give quantitative values to "the probability it went through slit 1" vs. "the probability that we don't know"? I see below that you define what these would mean in the specific case of the setup of the delayed choice quantum eraser, but as I note below I don't see how this could be generalized to any arbitrary setup.

In a sense we already have that with the delayed choice experiment if we set up a D4 detector symmetrically to D3 (why isn't it in there?).

Well, note that the paper on the delayed choice quantum eraser that I referred to does have a D4 symmetric to D3 in the schematic Fig. 1, though not in Fig. 2, I guess they didn't think any interesting new information would come from it so they didn't bother.

Then we have a 25% chance of detection of the idler at each of D1-4, so that means we have a 50% chance of not knowing which slit the signal photon went through, a 25% chance we'll know it was slit A, and a 25% chance we'll know it was slit B. I think we have those probabilities in any interpretation.

OK, but this is only for the special case where each detector is positioned in such a way as to 100% erase or 100% preserve the which-path information (giving either a perfect interference pattern or a perfect non-interference pattern), it doesn't help at all with the question of quantifying "partial information" if you have a detector somewhere else where this isn't true, that was the case my question was meant to discuss, sorry if it wasn't clear.

 

[Ken G]

So I'm suggesting something similar for "which path information" in a Bohmian interpretation, where what you do know (the detection of the idler at some detector) gives you information about something you don't know, but which is assumed to have an objective value (which slit the signal photon went through), and where you can calculate theoretically the set of different trajectories which are consistent with the fact you do know about the idler's detection.

Here's my problem with the whole "Bohmian trajectory" concept. For a trajectory to represent new information, it has to be predictive. Otherwise a trajectory is just a semantic label for a bunch of information we already have. If I say "I know the photon went through the left slit, so I can predict it will hit the left detector", then we have information content in that trajectory concept. If we say "I know the photon hit the left detector, and I'm calling that (via Bohmian trajectories or any way you like) a photon that went through the left slit", then I call that trajectory concept a longwinded label for the same photon, containing no actual new information.

So much for the information content of the Bohmian approach, what about the determinism issue? After all, that's the main motivation. If I can imagine a definite trajectory, then I can connect the left and right sides of that trajectory, and say nothing happened in between that breaks the determinism. But do I really have a deterministic process? Not really, because nobody who isn't using a deterministic approach (say, CI) ever claimed that the two-slit apparatus broke what was otherwise a deterministic process-- they would have said the process was not deterministic from the get go. So plotting trajectories through the apparatus completely begs the issue of whether or not this is actually a deterministic process. If a Bohmian says "the photon followed this path because it hit the detector here", I ask "but what determined that the photon would follow that path in the first place?" There's a lack of new information there, the Bohmian trajectories are going to start out with some kind of ergodic assumption at the input boundary, which is a stochastic assumption in the first place. You cannot argue a process is deterministic because there's no break in determinism, you have to argue that it starts out determined and also there is no break in the determinism. No Bohmian calculations ever actually do the latter, only the former.

So I claim the Bohmian approach has limited meaning for two reasons:
1) Bohm trajectories are never predictive, so they are in effect just longer-winded semantic labels for whatever information has already been established by the apparatus, and
2) Bohm trajectories do not represent a deterministic process, they only represent a process which does not alter its deterministic or nondeterministic status in the course of the development of the envisaged trajectories. Since other interpretations hold that no part of the process is strictly deterministic, providing a way to maintain that status during propagation is of no significance.
So in all, what Bohm accomplishes is a way to let you continue believing the process is deterministic if you already wanted to believe that for other reasons, but it offers no evidence that the process actually is deterministic, nor does it help you gain information that is dynamically active.

The reason I discussed this is that SpectraCat had trouble understanding how we could still talk about the which-path information being "erased" under the Bohmian interpretation, since in the Bohmian interpretation even when you get an interference pattern on the screen you can still tell which slit the signal photon went through based on whether it's on the upper or lower half of the screen.

I see what you are saying about the meaning of "erasing" the information, but note that when information is erased, it is real information-- the dynamically active information, and it really is erased. The semantic label that the Bohmians attach to photons that hit the left side of the detector cannot be erased because it isn't really information at all, it's just a longer label for the same information and it presents nothing testable or dynamically active about those photons. A Bohmian trajectory could pass around Mars for all the difference it would make-- it's just a label attached to the photons.

"Mixture of a mixed state and a superposition state" doesn't seem to make sense, wouldn't that just be another mixed state?

Yes, but as I said above, I believe it would be a particularly unusual type of mixed state-- one with nondiagonal elements (if "superposition state" carries the implication of including more than one eigenfunction of the observable, with phase information not normally present in a mixed state, as would be appropriate for weak measurements).

Are you just thinking out loud or have you seen such a "mixture" defined in a mathematical sense somewhere? In any case I didn't mean anything like that by "partial" information, I just meant putting the detector at some position midway between a position where you completely lose the which-path information (the position of D1 on the last page of the DCQE paper for example) and a position where you completely preserve the which-path information (the position of D3 for example).

And I'm saying that's exactly the kind of partial information I'm talking about. The eigenvalues are which slit, and if you get partial information about that, to whatever extent you decohere which slit, you'll get a mixed state of the which-slit eigenfunctions, and to whatever extent you don't decohere those eigenstates, you'll still have the superposition of both slits. When you go to make the density matrix for that state, you'll get diagonal contributions that look like the which-slit information, and you'll get off-diagonal elements that reflect the coherences you did not decohere with your detector placements in the which-slit basis.

In that case there might be a way to quantify a "partial" which-path information,

The trace of the density matrix?

and the amount of partial which-path information might correspond to the degree to which the coincidence count resembled the D0/D1 coincidence count (fig. 3) vs. the D0/D3 coincidence count (fig. 5).

No question, I would say.

My suggestion was that the Bohmian interpretation might give a nice way to quantify "partial information" since for every idler detected there will be a definite truth about which path the signal photon took, so you can calculate theoretically the proportion of photons in the coincidence count that went through one slit vs. the other (then the idea would be that the closer they are to evenly distributed between slits the closer you are to zero which-path information, and the closer they are to all having come through one of the two slits the closer you are to 1 bit of which-path information)

And again I believe that would all just be another semantic relabeling of the exact same information contained in the density matrix of the joint wavefunction of both photons, subjecting it to the same criticism I leveled above for the one-photon situation. It gets complicated with the entanglement, but I'd happily bet that once again all we get from the Bohm approach is longwinded trajectory-sounding labels for the same information we get in CI. Bohm mechanics isn't mechanics at all, it's language, which is fine if you like that language, but we should not be fooled that there is any different information content.

OK, but this is only for the special case where each detector is positioned in such a way as to 100% erase or 100% preserve the which-path information (giving either a perfect interference pattern or a perfect non-interference pattern), it doesn't help at all with the question of quantifying "partial information" if you have a detector somewhere else where this isn't true, that was the case my question was meant to discuss, sorry if it wasn't clear.

I don't agree, that just happened to be an easy example I gave. If the detectors are set up to give partial information, like beamsplitters that split 75% - 25% or some such thing, then you can still calculate a probability of which slit the signal photon went through if the idler is detected in the non-erasing detectors, and the fraction of the time that the idler is detected in the partially erasing detectors, you have a problem you can use classical methods to determine. An erasing detection will look like a target-shaped interference pattern on the detector, and a non-erasing detection will look like a simple beam on the detector. Maxwell's equations could do it, or so it seems to me anyway.

 

[DevilsAvocado]

... In the delayed choice quantum eraser paper, if you look at the D0/D1 coincidence pattern in Fig. 3 on p. 4, in the Bohmian interpretation would it true here (as with the normal double-slit interference pattern) that half the signal photons making this pattern went through the left slit while half went through the right? And for the D0/D3 coincidence pattern in Fig. 5, in the Bohmian interpretation would it be true that all the signal photons making this pattern went through the same slit?

Ehh... maybe for me and any other layman out there, let’s recap and state exactly what we are talking about. This colored image is easier to follow:

• Single (laser pumped) photons from the double-slit are entangled in the BBO crystal.
  
  
• The five (black) detectors are labeled D₀, D₁, D₂, D₃, and D₄.
  
  
• The three (green) beam splitters (50% thru) are labeled BS_a, BS_b, and BS_c. The two (gray) mirrors (100% reflection) are labeled M_a and M_b.
  
  
• The upper 'red slit' going thru the BBO is called B, and photons from this slit are marked with red lines.
  
  
• The lower 'light blue slit' going thru the BBO is called A, and photons from this slit are marked with light blue lines.
  
  
• The "signal" photon (either from A or B) passes the (yellow) Lens to go to detector D₀.
  
  
• The "idler" photon (either from A or B) is deflected by a prism PS that sends it along divergent paths depending on whether it came from slit A or slit B.
  
  
• The detection time of the signal photon is 8ns earlier than that of the idler.

Therefore AFAIK, the signal photons are ALWAYS 50% A and B, no matter what coincidence pattern you are looking at. Or did I miss something (again)??

AFAIK, the 'magic' in the Delayed choice quantum eraser experiment is that there is NO way for the signal photons at detector D₀ to 'know' – in advance – what will happen with the idler photons at detectors D₁, D₂, D₃, and D₄ – whether there will be a 'definite path knowledge' or not ...??

That means that NO interference pattern could be observed in the corresponding subset of signal photons at detector D₀, IF the idler photon detection was made at detectors D₃ or D₄ – but detector D₀ doesn’t have this information at the time of detection...

That is weird.

I made a very unscientific 'investigation' and compared R₀₂ & R₀₃ in http://arxiv.org/abs/quant-ph/9903047" (Fig. 4 & 5) by putting them on top of each other and made one 50% transparent (+ inverted colors). Unfortunate, there seems to be a difference in the scale of the two pictures, I stretched width & height to get as good match as possible.

My thought was that I should be able to figure out which data (points) comes from the signal photons at detector D₀, to get a better chance to see what really happens here... (please don’t laugh )

Either I did something wrong, or I’m missing something fundamental about this 'data processing' – because there are only two (2) matching data points in R₀₂ & R₀₃ ...?? They are marked with red circles in the picture below:

One would expect it to be much more matching 'signal data points' from D₀... 50% or something like that...

What did I miss?? What is wrong?? And why did the authors not make it much easier to 'filter out' signal and idler in the diagrams of "joint detection" rate (by different symbols)??

That would have been extremely interesting!

Clues anyone...?

 

[DevilsAvocado]

... The group velocity is given by taking the derivative of the frequency with respect to the inverse-wavelength, so here that velocity comes out p/m.

Thanks a lot!

 

[SpectraCat]

Ehh... maybe for me and any other layman out there, let’s recap and state exactly what we are talking about. This colored image is easier to follow: [snip]

Clues anyone...?

Rather than answer your specific questions .. which were a little hard to follow, I will suggest that you take a look at Cthugha's analysis in post #8 of this thread: https://www.physicsforums.com/showthread.php?t=320334. I found it extremely helpful to dispel my confusion and understand in detail why there is really nothing mystical going on with the DCQE .. aside from the run-of-the-mill mysticism of entanglement, that is.

The upshot is that the coincidence counting simply selects different sub-sets of the entangled pairs, taking advantage of the well-defined phase-relationships between the entangled signal and idler photons. If you look at the sub-sets where the idler has encountered the "eraser", then you recover the interference, otherwise you don't. In any case, there is NEVER an interference pattern for the single-photon detection events at the movable D₀ detector ... that is because the photons arriving there are polarization-tagged so that you always have which path information.

Anyway, if you read through that thread, and perhaps this more recent one: https://www.physicsforums.com/showthread.php?t=503667

it may help alleviate some of your confusion. I know I felt like I had a much deeper understanding after working through Cthugha's analysis.

 

[DevilsAvocado]

Rather than answer your specific questions .. which were a little hard to follow, ...

Thanks for the links, I’ll check it out ASAP!

I kinda had a feeling that my "explanation" didn’t make any sense... but please tell me the signal photons are ALWAYS 50% A and B, right?

... or it’s time for me to start 'feeding the birds' or any other 'high-tech activity' ...

 

[Ken G]

Therefore AFAIK, the signal photons are ALWAYS 50% A and B, no matter what coincidence pattern you are looking at. Or did I miss something (again)??

They are always symmetric in regard to A and B, but it's not always clear what 50% one or the other means-- I'd probably reserve that language for when which-way information is obtainable from the idler photon, because that sounds like "mixed state" language to me.

AFAIK, the 'magic' in the Delayed choice quantum eraser experiment is that there is NO way for the signal photons at detector D₀ to 'know' – in advance – what will happen with the idler photons at detectors D₁, D₂, D₃, and D₄ – whether there will be a 'definite path knowledge' or not ...??

I was going to go into my standard rant about how overblown the weirdness of quantum erasure experiments gets by the faulty language that is often used to describe them, but apparently this was all well explained in another thread, and summarized by SpectraCat here. Do follow that summary, it is important to understand what is actually weird here, and what actually isn't. For one thing, it should become clear to you why the relative timing between the idler and signal detection is completely irrelevant, and there is actually nothing strange in that fact. What is strange is the gymnastics that the detection patterns go through to make sure that the raw data at D0 presents us with no clue as to what happened to the idler photon, but the coincidence data is magically able to look like the raw data when which-way information is contained in the idler detection, but slices itself into two interference patterns when which-way information is erased in the idler detection. Exactly how it pulls that stunt is pretty weird, but it's need to do it is basic quantum mechanics, with no causality implications at all. As I said to JesseM above, I feel that the answer to "how it does it" has a perfectly classical analog, obtainable with Maxwell's equations. Indeed, the correspondence principle simply requires that be true.

 

[JesseM]

Therefore AFAIK, the signal photons are ALWAYS 50% A and B, no matter what coincidence pattern you are looking at. Or did I miss something (again)??

No, look at D3--if the idler goes to D3, that's only possible for a red path, which means the signal photon can only have gone through slit B (upper slit on the diagram). Likewise D4 is only possible for a blue path, which means the signal photon can only have gone through slit A.

That means that NO interference pattern could be observed in the corresponding subset of signal photons at detector D₀, IF the idler photon detection was made at detectors D₃ or D₄ – but detector D₀ doesn’t have this information at the time of detection...

No interference pattern is ever seen in the total pattern of signal photons at D0, only in various subsets, subsets which can only be considered after you know where the corresponding idler went--see [post=2840648]this post[/post] of mine for a discussion. So the signal photons don't need to anticipate anything, you're free to pick an interpretation where the detection of each signal photon changes the probability that the idler will go to different detectors, for example if a signal photon is detected near a peak of the D0/D1 coincidence count but near a valley of the D0/D2 coincidence count, that could increase the probability that the idler will go to D1 and decrease the probability the idler will go to D2.

I made a very unscientific 'investigation' and compared R₀₂ & R₀₃ in http://arxiv.org/abs/quant-ph/9903047" (Fig. 4 & 5) by putting them on top of each other and made one 50% transparent (+ inverted colors). Unfortunate, there seems to be a difference in the scale of the two pictures, I stretched width & height to get as good match as possible.

My thought was that I should be able to figure out which data (points) comes from the signal photons at detector D₀, to get a better chance to see what really happens here... (please don’t laugh )

All of the data points in those graphs come from D0. The blue dots represent signal photons at D0 whose corresponding idlers were detected at D2, while the white dots represent signal photons at D0 whose corresponding idlers were detected at D3. Remember, each signal photon is a member of an entangled pair, so for each signal photon detected, you can ask where that photon's idler "twin" was detected.

Either I did something wrong, or I’m missing something fundamental about this 'data processing' – because there are only two (2) matching data points in R₀₂ & R₀₃ ...?? They are marked with red circles in the picture below:

No, those aren't "matching data points", they're just positions that happened to have the same number of signal photons in both coincidence counts. Keep in mind those dots aren't individual photons, rather each dot represents the number of photons found at a particular position on the x-axis at D0. It might be clearer if these interference patterns were drawn as bar graphs, where each "hit" of a photon at a particular position increases the height of the bar at that position--see the animation http://www.cabrillo.edu/~jmccullough/Applets/Flash/Modern%20Physics%20and%20Relativity/DoubleSlitElectrons.swf.

What did I miss?? What is wrong?? And why did the authors not make it much easier to 'filter out' signal and idler in the diagrams of "joint detection" rate (by different symbols)??

Again the graphs don't show idlers at all, only numbers of signal photons at various positions along the x-axis for D0--but each graph deals only with a subset of signal photons whose idler "twins" went to a particular detector like D1 or D3. That's what's meant by a "coincidence count".

 

[Ken G]

Yes, the point matching must be just happenstance. Presumably it happens at the peaks since that's where most of the signal is the same in both slices of the data.

 

[unusualname]

As I said to JesseM above, I feel that the answer to "how it does it" has a perfectly classical analog, obtainable with Maxwell's equations. Indeed, the correspondence principle simply requires that be true.

Yeah, but earlier in the week you thought the measurement of the QM Wave Function was classical too, and were even suggesting two-photon entangled states might have a classical energy flow.

Please do show us a classical analog to delayed choice and explainable by Maxwell Equations.

 

[Ken G]

Yeah, but earlier in the week you thought the measurement of the QM Wave Function was classical too, and were even suggesting two-photon entangled states might have a classical energy flow.

Nothing I said earlier in the week was contradicted by anything that ensued. Nor did I ever say any of the things you just mentioned. What I actually said is a matter of record, and you'll find that it was that there's no evidence in any of these papers or threads on the papers that the "average trajectory" construct, made from aggregates of weak measurements, is any different from the classical concept of a Poynting flux streamline. That continues to be true, by the way, as those were all single-photon experiments, not entanglements.

Please do show us a classical analog to delayed choice and explainable by Maxwell Equations.

Entanglement complicates matters because correlation functions are involved. I'd be pretty shocked if it hasn't already been done, it seems like a pretty obvious application of the correspondence principle to the quantum erasure experiments. (And weren't you the one who found that Prosser did this for the single-photon case in the 1970s? Why would you doubt it's been done for quantum erasure by now?) Since you were able to so completely misconstrue what I said in the case of single photons, I haven't much hope of communicating this point in the entangled context. Go back to the single-particle threads and get the point there, first.

 

[unusualname]

Nothing I said earlier in the week was contradicted by anything that ensued. Nor did I ever say any of the things you just mentioned. What I actually said is a matter of record, and you'll find that it was that there's no evidence in any of these papers or threads on the papers that the "average trajectory" construct, made from aggregates of weak measurements, is any different from the classical concept of a Poynting flux streamline. That continues to be true, by the way, as those were all single-photon experiments, not entanglements.
Entanglement complicates matters because correlation functions are involved. I'd be pretty shocked if it hasn't already been done, it seems like a pretty obvious application of the correspondence principle to the quantum erasure experiments. (And weren't you the one who found that Prosser did this for the single-photon case in the 1970s? Why would you doubt it's been done for quantum erasure by now?) Since you were able to so completely misconstrue what I said in the case of single photons, I haven't much hope of communicating this point in the entangled context. Go back to the single-particle threads and get the point there, first.

Prosser found a result pretty much already found by Lorentz in 1898 or so, but Prosser had a computer to enable calculating trajectories over a large range. Its only interest to me is that it was the only other paper (apart from a bohmian one) qualitatively reproducing the "trajectories" found in the recent Science paper.

And you appeared to suggest to me that even a two-photon experiment involving weak measurements would produce some "trajectories" that probably had a classical flow, but you needed to see experimental details (fair enough, but I will just tell you it won't be the case)

And there need not be any entanglement in delayed choice experiments, eg the one I have recently pointed out in another thread Experimental realization of Wheeler's delayed-choice GedankenExperiment , but I can guarantee you'll never explain it with Maxwell's Equations.

No, I'm afraid delayed choice experiments to kill naive/intuitive/classical ideas about nature, and have no explanation using naive/intuitive/classical ideas despite what many people on this forum seem to think.

 

[JesseM]

Here's my problem with the whole "Bohmian trajectory" concept. For a trajectory to represent new information, it has to be predictive.

You're imposing limits on "information" that have nothing to do with information theory, there's no rule in information theory that says "new information" must be "predictive". And what about my discussion of macrostates and microstates in statistical mechanics? Would you say "for a microstate to represent new information, it has to be predictive"? But surely it's totally impossible in practice to ever measure the microstate of a macroscopic system involving vast numbers of particles, like a box of gas.

Otherwise a trajectory is just a semantic label for a bunch of information we already have. If I say "I know the photon went through the left slit, so I can predict it will hit the left detector", then we have information content in that trajectory concept. If we say "I know the photon hit the left detector, and I'm calling that (via Bohmian trajectories or any way you like) a photon that went through the left slit", then I call that trajectory concept a longwinded label for the same photon, containing no actual new information.

I think it's true that if you know the exact location the photon hit the screen, you can retroactively assign a precise Bohmian trajectory to it. That means that if you assume Bohmian mechanics is the correct hidden-variables theory, then in that context the Bohmian trajectory contains no new information beyond the information about the position on the screen where the photon landed. Similarly in any deterministic theory, if you know the state of an isolated system at some time T, you gain no additional information by learning its state at some earlier or later time assuming the system remains isolated for the entire interval. But, how is this in any way relevant to my argument? I was talking about the "information" you get about the path if you only know about where the idler was detected, and don't know anything about where the signal photon was detected on the screen, therefore you don't know enough to determine the Bohmian path even assuming that Bohmian mechanics is correct.

So much for the information content of the Bohmian approach, what about the determinism issue? After all, that's the main motivation. If I can imagine a definite trajectory, then I can connect the left and right sides of that trajectory, and say nothing happened in between that breaks the determinism. But do I really have a deterministic process? Not really, because nobody who isn't using a deterministic approach (say, CI) ever claimed that the two-slit apparatus broke what was otherwise a deterministic process-- they would have said the process was not deterministic from the get go. So plotting trajectories through the apparatus completely begs the issue of whether or not this is actually a deterministic process.

Again this seems to have nothing to do with what I (or anyone else) is arguing--is anyone using this experiment to try to "prove" that Bohmian mechanics is superior to the CI, or that determinism is true while indeterminism is false? Certainly I wasn't, I was just making the point that if you assume for the sake of argument that Bohmian mechanics is correct, then even in this case it still makes sense to talk about whether you do or don't gain any which-path information depending on which detector the idler is seen at (in response to SpectraCat's question about whether this terminology would make sense in a Bohmian context). So of course I have "begged the issue" of Bohmian mechanics and determinism being correct, but that's because I never intended to make any actual argument that they are correct in reality, just to explore what it would mean to talk about "which-path information" in a purely hypothetical world where they were correct.

If a Bohmian says "the photon followed this path because it hit the detector here", I ask "but what determined that the photon would follow that path in the first place?" There's a lack of new information there, the Bohmian trajectories are going to start out with some kind of ergodic assumption at the input boundary, which is a stochastic assumption in the first place.

The same sort of assumption about initial conditions is made in classical statistical mechanics (all microstates compatible with the initial macrostate , but that doesn't mean that classical statistical mechanics is nondeterministic. You can just take it as an application of the principle of indifference, for example.

So I claim the Bohmian approach has limited meaning for two reasons:
1) Bohm trajectories are never predictive, so they are in effect just longer-winded semantic labels for whatever information has already been established by the apparatus

But that's the whole idea of an "interpretation" of quantum mechanics--it's called an interpretation rather than a theory because it makes no new predictions, it's just a different ontological view of what's "really going on" behind the scenes.

2) Bohm trajectories do not represent a deterministic process, they only represent a process which does not alter its deterministic or nondeterministic status in the course of the development of the envisaged trajectories.

But broadly speaking, a Bohmian would say that the "course of the development of the envisaged trajectories" would encompass the entire history of the universe, there aren't assumed to be any "breaks" due to measurement or anything else (see the discussion of measurement in sections 7 and 8 here).

So in all, what Bohm accomplishes is a way to let you continue believing the process is deterministic if you already wanted to believe that for other reasons, but it offers no evidence that the process actually is deterministic

Sure, along with some other properties people may want to believe, like the idea that particles actually have well-defined values for classical properties like position and velocity at all times. No one claims there is any evidence that Bohmian mechanics is true AFAIK, it's just an interpretation, one of many.

"Mixture of a mixed state and a superposition state" doesn't seem to make sense, wouldn't that just be another mixed state?

Yes, but as I said above, I believe it would be a particularly unusual type of mixed state-- one with nondiagonal elements (if "superposition state" carries the implication of including more than one eigenfunction of the observable, with phase information not normally present in a mixed state, as would be appropriate for weak measurements).

Why is that unusual? I haven't studied density matrix formalism in any detail but my basic understanding (see discussion in [post=3245596]this post[/post]) is that it's common for there to be off-diagonal elements, although if you're working in the position basis then decoherence causes them to become close to zero fairly quickly.

Anyway, what density matrix are you talking about? Are you talking about the "reduced density matrix" for a single member of the entangled two-particle system? It seems to me that if you were dealing with the full state of the two-particle system there'd be no need for a density matrix, this system is in a pure state prior to measurement and measurement of one member simply collapses it to a new pure state. If you're talking about the reduced density matrix for just the signal photon, what basis are you assuming? And would you be talking about the reduced density matrix for the signal photon after we know the idler has been found at a particular detector, or are you maybe assuming ordinary classical uncertainty about which detector the idler was seen at? (i.e. a 25% chance it was at any of the four detectors D1-D4)

And I'm saying that's exactly the kind of partial information I'm talking about. The eigenvalues are which slit

How can the eigenvalues be which slit, given that you don't measure the position of the signal photon at the slits? I suppose if you know the exact time the signal photon would be measured to be passing through the slits, then if the idler has already been detected at D3 or D4, in either of those cases there would be a probability 1 the signal photon would be detected in the slit corresponding to that detector at that exact time. But at later times the position of the photon won't be at the slits at all.

Are you imagining a "which-slit" operator different from the position operator? I don't really see how that would work, so if you are talking about something like that can you express the eigenstates of this operator as weighted sums of eigenstates of some other known operator?

In that case there might be a way to quantify a "partial" which-path information,

The trace of the density matrix?

Can you explain why you say that? What density matrix (again, a reduced density matrix for the signal photon or a density matrix for the 2-particle system based on classical uncertainty about which detector the idler goes to?), and in what basis? Would the trace be zero in the specific case of the idler being detected at D1 or D2, and one (one bit, corresponding to knowledge of which of the two slits the photon went through) in the case of the idler being detected at D3 or D4?

and the amount of partial which-path information might correspond to the degree to which the coincidence count resembled the D0/D1 coincidence count (fig. 3) vs. the D0/D3 coincidence count (fig. 5).

No question, I would say.

I don't there's a good basis for that level of confidence--neither of us has given a precise mathematical definition of "partial which-path information" that can be applied even in cases where the detectors are placed at other positions (say, midway between D1 and D3 in the standard DCQE experiment), much less proven that there would be a one-to-one relationship between "amount of partial which-path information" and "amount of interference in coincidence count". It seems intuitively plausible based on intuitions about complementarity, but intuitions are often misleading in physics.

My suggestion was that the Bohmian interpretation might give a nice way to quantify "partial information" since for every idler detected there will be a definite truth about which path the signal photon took, so you can calculate theoretically the proportion of photons in the coincidence count that went through one slit vs. the other (then the idea would be that the closer they are to evenly distributed between slits the closer you are to zero which-path information, and the closer they are to all having come through one of the two slits the closer you are to 1 bit of which-path information)

And again I believe that would all just be another semantic relabeling of the exact same information contained in the density matrix of the joint wavefunction of both photons

I'm not claiming there is any new information in the Bohmian paths beyond what you'd have if you knew the precise position of the signal photon hitting the screen. I'm just saying that in the ordinary version of QM, there's no obvious way to go about choosing a definition of "partial which-path information", since information is ordinarily understood in terms of classical probabilities but QM doesn't allow you to talk about the "probability" the photon went through one slit or the other in cases where you didn't actually measure which slit it went through. Bohmian mechanics does, so it might be a good start if we were looking find the "right" definition of "partial which-path information", the one which we hope will map directly to "amount of interference". Once you have already defined what "partial which-path information" is supposed to mean mathematically, of course you could dispense entirely with the Bohmian interpretation and just use this definition in the context of any other interpretation of QM including CI, seeing it as just a nice feature of this definition that if you have X bits of partial which-path information, then X also equals the Shannon entropy based on the probabilities the signal photon went through one slit or the other in the Bohmian interpretation, but certainly not saying that this is some sort of proof that Bohmian mechanics is correct while other interpretations are wrong. But of course this all depends on the assumption that a definition of partial which-path information like this would even be useful insofar as its value would directly correspond to the amount of interference in the coincidence count, which is just a speculation on my part that could easily be wrong.

 

[Ken G]

Prosser found a result pretty much already found by Lorentz in 1898 or so, but Prosser had a computer to enable calculating trajectories over a large range. Its only interest to me is that it was the only other paper (apart from a bohmian one) qualitatively reproducing the "trajectories" found in the recent Science paper.

All true. But I'm not sure you are getting the punchline-- to the extent that the Prosser results are the same as the average trajectory figure, the latter contains nothing but classical information, regardless of how it is made. I'm saying the same could be done for an erasure experiment in the classical limit of correlations of bright radiation fields, that's just the correspondence principle. Doing that would give, as I said, the "classical analog" of the answer to what is happening that let's the photons pull these remarkable stunts in quantum erasure experiments.

And you appeared to suggest to me that even a two-photon experiment involving weak measurements would produce some "trajectories" that probably had a classical flow, but you needed to see experimental details (fair enough, but I will just tell you it won't be the case)

When you understand what I'm saying, you'll understand why it will have to be the case. Unless you think the correspondence principle is wrong, that is.

And there need not be any entanglement in delayed choice experiments, eg the one I have recently pointed out in another thread Experimental realization of Wheeler's delayed-choice GedankenExperiment , but I can guarantee you'll never explain it with Maxwell's Equations.

That's an interesting experiment, but I don't think it shows what you think it shows. If one adopts a standard QM interpretation, say CI, there is nothing that happens in that GedankenExperiment that is the least bit surprising. One simply evolves the wave packet in time, and if you make your choice to open or close the final beamsplitter before the wavepacket arrives there (which is what you are in fact doing), it's not surprising that the choice affects the outcome of the experiment, regardless of whether you do it after the wave packet "enters the interferometer", which is irrelevant.

The real point of that experiment is it shows you why you need a wavefunction if you want to interpret causality in the mundane way. If you instead choose a Bohmian approach, where the trajectories really mean something when the wave packet first enters the interferometer, only then do you have a problem with causality-- only then do you have to worry about how the trajectories know which way to go before the decision to open or close is made. The Bohmians have to invoke a non-causal pilot wave to account for this, as I understand it, and that's a bit of a headache. But standard quantum mechanics just time-evolves the wavefunction, and nothing subtle at all is happening with causality-- you just have to let go of the trajectory concept, which is pretty much a key lesson of standard quantum mechanics anyway The experiment is an excellent way to make this point, so thank you for citing it, but I don't think it makes any of the points you seem to be stressing.

No, I'm afraid delayed choice experiments to kill naive/intuitive/classical ideas about nature, and have no explanation using naive/intuitive/classical ideas despite what many people on this forum seem to think.

I suggest it is your own impression of what those people think that is at fault here-- you just haven't understood them. Your reframing of what I think demonstrates quite clearly that you have not understood what I think.

 

[unusualname]

@KenG

I think your understanding of QM is excellent, you explain loads of stuff here with superb clarity (better than I could)

I can only think you have got a bit muddled trying to denounce Bohmian ideas or whatever that have made some of your posts a little confusing.

 

[Ken G]

You're imposing limits on "information" that have nothing to do with information theory, there's no rule in information theory that says "new information" must be "predictive".

If so, then give me an example of something that you consider to be information, and I will tell you why that information is in fact predictive. I don't think you are mistaken about what information is, I think you have not recognized its ramifications.

And what about my discussion of macrostates and microstates in statistical mechanics? Would you say "for a microstate to represent new information, it has to be predictive"?

Absolutely yes. I completely understood your clear description of information in microstates, and it is fully consistent with everything I said. Here's how to test this-- imagine every particle, in addition to "spin", has "angels". Let's attribute 105 angel states to each particle. Now suddenly we have all these new microstates, all this new information. But wait, if the angels are dynamically inert, none of the old predictions will be altered one iota by their introduction into the microstates. We could indeed associate angels with the number of angels that can fit on a pin if we want. Is there any information there? No there isn't-- dynamically ignorable information is not information at all, because information is not some kind of ontological statement about reality (if it were, physics wouldn't work until we'd found all the possible angel modes of information out there), it is simply what we need to know about the system to predict its behavior. I'm sure you'll find the same thing in information theory, and game theory for that matter.

What I'm saying about Bohmian trajectories is that the illusory information we might imagine is contained in them is actually quite dynamically inert, which is the same thing as saying it is non-predictive, which is the same thing as saying it isn't information at all. Counting microstates is information only if it matters to the outcome-- only if those microstates aren't angels, but are instead dynamically active, and predictively important. This would seem to be the key issue we need to resolve.

Why is that unusual? I haven't studied density matrix formalism in any detail but my basic understanding (see discussion in [post=3245596]this post[/post]) is that it's common for there to be off-diagonal elements, although if you're working in the position basis then decoherence causes them to become close to zero fairly quickly.

I'll comment on this. It's not new physics to have off-diagonal elements to density matrices, but that's not what you get in measurements. The whole difference between what we would call a strong measurement, and a weak one, is the absence of presence of off-diagonal elements of the resulting density matrix. So if you're saying there's nothing especially important there, then I'm saying exactly-- that's why there's not going to be any breakthrough importance to weak measurements in distinguishing Bohmian trajectories from other ways of making predictions. I've said many of the same things you just said-- interpretations are not predictive, they are philosophical, and this also means they do not monkey with any of the information content. But they can produce the illlusion of doing so, and that's exactly what I see happening as soon as people start talking about testing Bohmian trajectories with some new wrinkle in the apparatus.

Anyway, what density matrix are you talking about?

The projection of the unitary apparatus onto the quantum subspace eigenstates.

Are you talking about the "reduced density matrix" for a single member of the entangled two-particle system?

That depends on which quantum subspace you are projecting onto. Probably you don't want to project out the entanglement, so you'd rather look at the two-particle density matrix, even though it is a more sophisticated mathematical object.

It seems to me that if you were dealing with the full state of the two-particle system there'd be no need for a density matrix, this system is in a pure state prior to measurement and measurement of one member simply collapses it to a new pure state.

No, that would be nonunitary. You'll never get that from the equations, it is a manual post-processing step we do to start creating language about what happened. It's not what you want to do here.

If you're talking about the reduced density matrix for just the signal photon, what basis are you assuming?

The which-way basis is one good choice. It depends on what you are trying to do. My comments are general in nature.

How can the eigenvalues be which slit, given that you don't measure the position of the signal photon at the slits?

If there is which-way information, the eigenvalue is which slit. It doesn't matter how you get that information.

I suppose if you know the exact time the signal photon would be measured to be passing through the slits, then if the idler has already been detected at D3 or D4, in either of those cases there would be a probability 1 the signal photon would be detected in the slit corresponding to that detector at that exact time. But at later times the position of the photon won't be at the slits at all.

Making time measurements complicates the apparatus and changes the nature of the coherences being studied. It would seem to be better to stick to coincidence counting.

Are you imagining a "which-slit" operator different from the position operator?

Yes, it is just a shorthand for simplifying the full wavefunction. Nothing new there, we're already talking about discrete outcomes in a CCD, this is just even more discrete. All for simplicity sake, it's all the same guts.

I don't really see how that would work, so if you are talking about something like that can you express the eigenstates of this operator as weighted sums of eigenstates of some other known operator?

Why would I want to do that? It's a perfectly normal operator, with 2 eigenvalues, invoked any time anyone uses the phrase "which-way information". Information in quantum mechanics corresponds to eigenvalues of operators, it can't come from anything else.

Can you explain why you say that? What density matrix (again, a reduced density matrix for the signal photon or a density matrix for the 2-particle system based on classical uncertainty about which detector the idler goes to?), and in what basis? Would the trace be zero in the specific case of the idler being detected at D1 or D2, and one (one bit, corresponding to knowledge of which of the two slits the photon went through) in the case of the idler being detected at D3 or D4?

The reduced density matrix for the signal quantum substate, in the which-way basis.

I'm not claiming there is any new information in the Bohmian paths beyond what you'd have if you knew the precise position of the signal photon hitting the screen. I'm just saying that in the ordinary version of QM, there's no obvious way to go about choosing a definition of "partial which-path information", since information is ordinarily understood in terms of classical probabilities but QM doesn't allow you to talk about the "probability" the photon went through one slit or the other in cases where you didn't actually measure which slit it went through. Bohmian mechanics does, so it might be a good start if we were looking find the "right" definition of "partial which-path information", the one which we hope will map directly to "amount of interference". /quote]No, that's exactly what I'm arguing Bohmian mechanics does not do, expressly because there is no additional information involved. If it empowered us to make statements about probabilities that CI did not, that would require additional information. Just what information do you see in a Bohmian trajectory that cannot be extracted in the CI interpretation, such that you could assert which-way probabilities that CI would be blind to? My point is that you will not have a good answer to this question, and that is the answer to your question.

Once you have already defined what "partial which-path information" is supposed to mean mathematically, of course you could dispense entirely with the Bohmian interpretation

Yes, this is exactly the point. There is no difficulty defining "partial which-path information" in a CI interpretation, and I claim there is even a classical analog, as I outlined above (but a more explicit calculation would be quite involved and certainly should have been done somewhere by now). It's all right there in the reduced density matrix of the photon in question, in the which-way basis. Give me an example of a setup and I'll tell you how the CI extracts partial which-way information from it. There's no different information in the Bohmian approach, expressly because there is no difference in predictive power. Bohmian trajectories are just a pictorially suggestive way of labeling the information that is already there in standard quantum mechanics, in the density matrices in the which-way basis (when we simplify Bohmian trajectories to the which-way basis, as this is the decisive feature for controlling the qualitative attributes of the full patterns in the x basis).

 

[JesseM]

If so, then give me an example of something that you consider to be information, and I will tell you why that information is in fact predictive. I don't think you are mistaken about what information is, I think you have not recognized its ramifications.

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. For example, if I subscribe to a religion that says every pin has a number of angels from 1-1000 dancing on it (with each number occurring equally frequently for the set of all pins), then a soothsayer holds up a pin and tells me he had a vision that told him the number of angels dancing on this pin is somewhere between 300 and 400 but that no human will ever learn the true number, then under the assumption that this model of angels on pins is correct, I have gained some additional information about the (unknown) "true" number, which could be quantified by a change in shannon entropy associated with the number of angels on that pin. But of course if I subscribe to a model where this religion is a lie, all I've learned is something about the soothsayer's delusions.

And what about my discussion of macrostates and microstates in statistical mechanics? Would you say "for a microstate to represent new information, it has to be predictive"?

Absolutely yes. I completely understood your clear description of information in microstates, and it is fully consistent with everything I said.

Really? But do you agree it's totally impossible in practice to determine the exact microstate of a box of gas? If so, then any quantification of the "information" about the microstate associated with knowledge of a given macrostate is obviously just as much based in belief in the model as the example with angels.

Here's how to test this-- imagine every particle, in addition to "spin", has "angels". Let's attribute 105 angel states to each particle. Now suddenly we have all these new microstates, all this new information. But wait, if the angels are dynamically inert, none of the old predictions will be altered one iota by their introduction into the microstates.

Well, you can have a classical thermodynamic model that predicts all the same things about the evolution of macro-variables like pressure and temperature as statistical mechanics, but without bothering to postulate any "microstates". 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). 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"? Again there is nothing in information theory to justify such a claim, so if you are saying this it's just a new rule you've made up (which you're free to do of course, but don't expect everyone else to follow it).

Is there any information there? No there isn't-- dynamically ignorable information is not information at all, because information is not some kind of ontological statement about reality (if it were, physics wouldn't work until we'd found all the possible angel modes of information out there)

That's a total non sequitur, since anyone who is interested in "interpretations" of QM which postulate additional aspects to the world besides measurable ones is naturally going to acknowledge that physics "works" in a predictive sense even if you ignore these additional aspects. 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?

I'm sure you'll find the same thing in information theory, and game theory for that matter.

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. 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).

I'll comment on this. It's not new physics to have off-diagonal elements to density matrices, but that's not what you get in measurements. The whole difference between what we would call a strong measurement, and a weak one, is the absence of presence of off-diagonal elements of the resulting density matrix.

You can have a density matrix in any basis you like, not necessarily the measurement basis.

I'm saying exactly-- that's why there's not going to be any breakthrough importance to weak measurements in distinguishing Bohmian trajectories from other ways of making predictions.

I already explained that I was never asserting anything like this, nor as far as I can see was anyone else on the thread, so why do you keep talking about this?

Anyway, what density matrix are you talking about?

The projection of the unitary apparatus onto the quantum subspace eigenstates.

What is "unitary apparatus"? Do you mean the pure state of apparatus + photons? If so why bother with the apparatus, why not just worry about the pure state of the photons themselves? And what subspace are you projecting on specifically? That was the whole point of my question "what density matrix".

That depends on which quantum subspace you are projecting onto. Probably you don't want to project out the entanglement, so you'd rather look at the two-particle density matrix, even though it is a more sophisticated mathematical object.

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? 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?

It seems to me that if you were dealing with the full state of the two-particle system there'd be no need for a density matrix, this system is in a pure state prior to measurement and measurement of one member simply collapses it to a new pure state.

No, that would be nonunitary.

Sure, I was thinking in terms of the standard method of doing calculations in QM, not some ontological statement about what the true dynamics of the state of the universe are. Obviously if you include the quantum state of the apparatus into the full quantum state, then you can explain the appearance of "collapse" in terms of decoherence which is still unitary, but in practice people don't normally do this when they just want to calculate probabilities a photon will be detected at a particular location.

It's not what you want to do here.

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.

If you're talking about the reduced density matrix for just the signal photon, what basis are you assuming?

The which-way basis is one good choice. It depends on what you are trying to do. My comments are general in nature.

I don't see "the which-way basis" as a meaningful answer, see below.

I don't really see how that would work, so if you are talking about something like that can you express the eigenstates of this operator as weighted sums of eigenstates of some other known operator?

Why would I want to do that? It's a perfectly normal operator, with 2 eigenvalues, invoked any time anyone uses the phrase "which-way information".

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. 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? 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. Likewise for the whole concept of "partial which-path information", if you don't have some clear method of calculating the value of that concept in any arbitrary situation (like after the idler has been detected at a position midway between D1 and D3), then this term has no well-defined meaning, and handwaving about the "which-way basis" won't help.

Information in quantum mechanics corresponds to eigenvalues of operators, it can't come from anything else.

Physically measurable information has to come from eigenvalues of operators since that's all we ever measure, but you can have measurable information about any arbitrary function of these eigenvalues. Presumably if there was a good way to define "partial which-path information" it could be defined as a function of the (arbitrary) position the idler was detected in a DCQE type setup, and that position is of course an eigenvalue.

I'm not claiming there is any new information in the Bohmian paths beyond what you'd have if you knew the precise position of the signal photon hitting the screen. I'm just saying that in the ordinary version of QM, there's no obvious way to go about choosing a definition of "partial which-path information", since information is ordinarily understood in terms of classical probabilities but QM doesn't allow you to talk about the "probability" the photon went through one slit or the other in cases where you didn't actually measure which slit it went through. Bohmian mechanics does, so it might be a good start if we were looking find the "right" definition of "partial which-path information", the one which we hope will map directly to "amount of interference".

No, that's exactly what I'm arguing Bohmian mechanics does not do, expressly because there is no additional information involved. If it empowered us to make statements about probabilities that CI did not, that would require additional information.

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.

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?

Just what information do you see in a Bohmian trajectory that cannot be extracted in the CI interpretation, such that you could assert which-way probabilities that CI would be blind to?

There are no measurable parameters that couldn't be used by both. But certainly the interpretation of a "complementarity parameter" could be different in the two interpretations, if it was the case that the value of this parameter always matched the shannon entropy of the probabilities the signal photon went through different slits in the Bohmian interpretation--i.e. if you detect the idler at position X, and that implies in Bohmian mechanics that there's a probability p1 the signal photon went through the left slit and a probability p2 it went through the right, then the value of the 'complementarity parameter' works out to -[p1*ln(p1) + p2*ln(p2)]. 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. But this is not some sort of weakness of the CI, any more than my inability to assign a meaningful probability to the number of angels on a pin is a weakness of my nonbelief in the every-pin-has-angels doctrine.

There is no difficulty defining "partial which-path information" in a CI interpretation

Does "no difficulty" just mean that if we had come up with some definition, perhaps using Bohmian intuitions, there would be no difficulty using this definition in the context of the CI as well? 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"? 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).

and I claim there is even a classical analog, as I outlined above (but a more explicit calculation would be quite involved and certainly should have been done somewhere by now).

Where in your post did you say anything about a "classical analog"? I don't see it.

It's all right there in the reduced density matrix of the photon in question, in the which-way basis.

And again, "which-way basis" is itself a completely undefined phrase that is of no use if what I know is the position X where the idler was detected, since if that position X is neither "wholly which-path erasing" or "wholly which-path preserving" I don't know how to translate the quantum state into the "which-way basis" in this case.

Give me an example of a setup and I'll tell you how the CI extracts partial which-way information from it.

OK, my example is almost like the DCQE, but now draw a straight line between D1 and D3, and put another detector D5 along this line, 1/4 away from D3 and 3/4 away from D1. If the idler is detected at D5, how do I figure out how much "partial which-way information" I learn from this?

 

[Ken G]

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.