A delayed choice experiment with a paradox. What did I do wrong?

In summary, the conversation discusses a gedanken experiment involving a down converter crystal that emits two photons when hit by one. The experiment includes half-silvered mirrors and detectors to observe interference patterns. The speakers discuss the possibility of obtaining "which-way information" about the original photon and its effect on the interference pattern. They also mention the concept of retro-causation and the violation of quantum mechanics principles. The conversation ends with one speaker searching for more information on the topic.
  • #1
Fredrik
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Some time ago I read some stuff about delayed choice and quantum eraser experiments, but rather than reading the articles thorouhgly, I just glanced at them quickly, picked up a few ideas, and then tried to design a gedanken experiment of my own. Now I seem to have run into a bit of a problem. My gedanken experiment seems to either contain a paradox or violate the principles of quantum mechanics. I'm sure I'm making some sort of mistake, but so far I haven't been able to find it. Perhaps someone here can can help me figure this out.

Look at the attached drawing. A "down converter" is a crystal that that emits two photons, each with half the energy of the original, when it's hit with just one. According to what I read, such crystals actually exist. I imagine that it would be difficult, if not practically impossible to build a down converter that always emits the two photons in the directions I've drawn in the picture, but I don't think it's impossible in principle.

Imagine that we have carefully adjusted the lengths of the paths in this experiment so that the lower right detector never clicks, because of destructive interference. All the photons that reach the half-silvered mirror below the detectors will go to the left detector.

If we imagine the same setup, but without the half-silvered mirror at the upper right, there wouldn't be any interference at the lower detectors. They would both detect the same number of photons. This is because the upper detectors would give us "which-way information" about the original photon. For example, the detector at the top of the picture will only click if the original photon was reflected at the first half-silvered mirror.

The purpose of the half-silvered mirror at the upper right (and the mirrors at the upper left and lower right), is to eliminate the possibility to obtain which-way information that would destroy the interference. There's no way, even in principle, to determine if a photon that's been detected by one of the detectors at the upper right went through the last half-silvered mirror or was reflected by it.

Let's move on to the paradox. Suppose that we sometimes remove the half-silvered mirror at the upper right, after one of the lower detectors has clicked. This will make it possible to determine which way the original photon went at the first half-silvered mirror, and this should destroy the interference at the lower two detectors. If the interference really is destroyed, then the lower right detector should detect a photon 50% of the times we do this (instead of 0%).

This is retro-causation. The effect precedes the cause. The detector will click because someone at a later time removed a half-silvered mirror. If that isn't strange enough, suppose that we immediately after the lower right detector has detected a photon decide not to remove the half-silvered mirror. Then we have a paradox. If we never remove the half-silvered mirror, that detector should never click.

Maybe it's just impossible to destroy the interference by removing the half-silvered mirror after one of the lower detectors has already clicked. But in that case, we can obtain which-way information about the original photon and still get interference. Doesn't this violate the principles of quantum mechanics?
 

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  • #2
I had to shrink the image to upload it. That made it rather unpleasant to look at. If you'd like to see a version of that picture that's a bit easier on the eyes, you can use http://web.comhem.se/~u87325397/weird_experiment.bmp [Broken].
 
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  • #3
I spent a little time today trying to get my head around this. I need to spend some more. :smile:
 
  • #4
Hello Fredrik,

I tried to follow your gedanken-experiment. It would be nice if you could give
a link to the article you read, so we'd have background knowledge and understand your idea better.


Fredrik said:
...
If we imagine the same setup, but without the half-silvered mirror at the upper right, there wouldn't be any interference at the lower detectors. They would both detect the same number of photons. This is because the upper detectors would give us "which-way information" about the original photon. For example, the detector at the top of the picture will only click if the original photon was reflected at the first half-silvered mirror.
...

I don't agree with this part. Even if you remove the H-mirror (right top)
and know which way the photon took at the first H-mirror, then, in my opinion, you would still get interference at the middle detectors.
For example:
Suppose we have no H-mirror (right top) and detect a photon at the right top
detector. This means: We know the photon was NOT reflected at the first H-mirror, but it went through it.
So, the photon goes straight through the first H-mirror, is downconverted,
and then is incident on the H-mirror in the center. Because this mirror in the center is half-silvered, you don't know which way it takes (in the center) and hence still get interference.

I think what you are considering has something to do with two-photon-interference, where you look if there's COINCIDENCE between the detectors in the center and on the top. Polarization plays a role there.
http://www.bu.edu/qil/pdf/PRL-09-02-96.pdf [Broken]
 
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  • #5
Edgardo said:
I tried to follow your gedanken-experiment. It would be nice if you could give
a link to the article you read, so we'd have background knowledge and understand your idea better.
Unfortunately I don't remember where I got most of the ideas. I know it started with an article by a guy named Ulrich Mohrhoff that I read about a year ago... I will do some googling when i get home today and see if I can find the articles I've read about similar experiments.

Edgardo said:
I don't agree with this part. Even if you remove the H-mirror (right top)
and know which way the photon took at the first H-mirror, then, in my opinion, you would still get interference at the middle detectors.
For example:
Suppose we have no H-mirror (right top) and detect a photon at the right top
detector. This means: We know the photon was NOT reflected at the first H-mirror, but it went through it.
So, the photon goes straight through the first H-mirror, is downconverted,
and then is incident on the H-mirror in the center. Because this mirror in the center is half-silvered, you don't know which way it takes (in the center) and hence still get interference.
I wouldn't call that interference. The interference I'm talking about will cause the left detector to click 100% of the time and the right detector to click 0% of the time (assuming that the path lengths have been carefully adjusted to make sure that happens). When things happen the way you describe, the left detector will click 50% of the time and the right 50% of the time. This happens because there's no interference between two different ways that a photon might reach the right detector (or the left detector).

Note that after the first half-silvered mirror, every line drawn in the diagram represents a photon that's in a superposition of existence and non-existence. In the language that's often used in discussions about similar experiments, each line represents a "half" photon. The interference I'm talking about is interference between two "halves" that have the last part of their paths to the detector in common. If we know that the original photon went through the first half-silvered mirror, then there was never any "half" photons around that could interfere with each other.

Edgardo said:
I think what you are considering has something to do with two-photon-interference, where you look if there's COINCIDENCE between the detectors in the center and on the top. Polarization plays a role there.
http://www.bu.edu/qil/pdf/PRL-09-02-96.pdf [Broken]
Thank you for your comments. I haven't had time to read that article yet. I will check it out later today.
 
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  • #6
caribou said:
I spent a little time today trying to get my head around this. I need to spend some more. :smile:
I appreciate comments like this too. It's good to know that someone is at least reading my post and thinking about it. Some people might choose not to read it, because it's rather long, but they don't know what they're missing. I believe that anyone who is even slightly interested in quantum mechanics will find this experiment very interesting.
 
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  • #7
Fredrik said:
I wouldn't call that interference. The interference I'm talking about will cause the left detector to click 100% of the time and the right detector to click 0% of the time (assuming that the path lengths have been carefully adjusted to make sure that happens). When things happen the way you describe, the left detector will click 50% of the time and the right 50% of the time. This happens because there's no interference between two different ways that a photon might reach the right detector (or the left detctor).

You're absolutely right. I wrote some bs there. I don't know how I got to that idea. The situation with 100% to click and 0% is the same as in the Mach-Zehnder-Interferometer http://www.upscale.utoronto.ca/GeneralInterest/Harrison/MachZehnder/MachZehnder.html
I have to admit that I've understood for the first time, why this 100% and 0% situation occurs, the site above really gives a good explanation.

I have to think about your gedankenexperiment.
 
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  • #8
Fredrik said:
Let's move on to the paradox. Suppose that we sometimes remove the half-silvered mirror at the upper right, after one of the lower detectors has clicked. This will make it possible to determine which way the original photon went at the first half-silvered mirror, and this should destroy the interference at the lower two detectors. If the interference really is destroyed, then the lower right detector should detect a photon 50% of the times we do this (instead of 0%).

Ok, I think I fully understood your paradox. My thought:
If you don't remove the H-mirror in the right top, only detector 1
in the center will click due to interference (like in the Mach-Zehnder-interferometer). I don't think that you can destroy
the interference afterwards. I can't imagine how that would look like.
So I agree with your comment:
Fredrik said:
Maybe it's just impossible to destroy the interference by removing the half-silvered mirror after one of the lower detectors has already clicked.

But on the other hand:

Fredrik said:
But in that case, we can obtain which-way information about the original photon and still get interference. Doesn't this violate the principles of quantum mechanics?

That is a good question. I'd say let the Gedankenexperiment become real
and test it in the lab.
:tongue:
 
  • #9
I think for the detectors in the right top the probabilities
should also be 100% and 0%, because the outer mirrors also
represent a Mach-Zehnder interferometer.
I just noticed that your experimental setup
consists of two Mach-Zehnders, an inner and an outer interferometer.
 
  • #10
Edgardo said:
I think for the detectors in the right top the probabilities should also be 100% and 0%
We can choose those percentages to be whatever we want them to be, simply by making the right path a little bit longer than the left path.
 
  • #11
It's an interesting addition to the experiments to have as Edgardo says, and I also noticed, essentially two interferometers in inner and outer loops. :smile:

I've been drawing the experiment as a superposition of two alternative histories which interfere.

After the first H mirror, one of my drawings is the history which follows the path of photon up and then the paths of the two photons created this way, while the other drawing is the history which follows path of the photon right and then the paths of the two photons created that way.

I'm imagining the drawings as then interfering and seeing if that gives me some clues.

I'd guess the photon goes around the inner loop by two paths in superposition has inteference effects with itself whatever else happens.

What is happening to the photon in the outer loop is what I'm wondering now.
 
  • #12
Fredrik-- the quote below greatly puzzles me.

If we imagine the same setup, but without the half-silvered mirror at the upper right, there wouldn't be any interference at the lower detectors. They would both detect the same number of photons. This is because the upper detectors would give us "which-way information" about the original photon. For example, the detector at the top of the picture will only click if the original photon was reflected at the first half-silvered mirror.

My instinct is to say "higher detectors" should replace "lower detectors" in the first sentence, there's a typo in other words.

But, perhaps not. In which case note that you have a 'sort-of" standard interferometer with the lower detectors. That is to say, interference will be alive and well at the lower detectors no matter the state of the upper apparatus. This is certainly true for classical E&M, and for QM as well. the Basic idea can be shown by looking at thr quantum fields at the measurement points. Both classical and quantum fields share the same spatial form. So at Lower Detector 1 (left), or LD1, have for 2 photons, with momentum k, a wave function (poor nomenclature)

EXP (-k{x11 + x21}) where x11 and x21 are the photon coordinates at LD1, fully symmetric in the two photons. Interference is demonstrated by varying x11 + x21. If the sum is pi/2 there is full destructive interference, and so on. Two channels feeding a detector guarantees interference. After the fact causality? I don't think so.

Regards,
Reilly Atkinson
 
  • #13
reilly said:
Fredrik-- the quote below greatly puzzles me.

If we imagine the same setup, but without the half-silvered mirror at the upper right, there wouldn't be any interference at the lower detectors. They would both detect the same number of photons. This is because the upper detectors would give us "which-way information" about the original photon. For example, the detector at the top of the picture will only click if the original photon was reflected at the first half-silvered mirror.

My instinct is to say "higher detectors" should replace "lower detectors" in the first sentence, there's a typo in other words.
Nope, that's not a typo. It's rather obvious that there will be no interference between "half" photons at the upper detectors without the half-silvered mirror at the upper right, but that's not the point. The point is that if that mirror is gone, the detector at the top of the picture can only click if the orignal photon was reflected by the half-silvered mirror near the bottom of the picture. If the original photon is reflected there, rather than going into a superposition of being reflected and passing through, then there can't be any interference at the lower detectors.


reilly said:
After the fact causality? I don't think so.
I'm not against the idea of "after the fact causality", but in this case it seems to lead to a paradox, and I'm against paradoxes. :smile:
 
  • #14
Hello Fredrik,

I've read about quantum erasers. And read these websites:
http://www.lns.cornell.edu/spr/1999-03/msg0015208.html

and especialy this one:
http://grad.physics.sunysb.edu/~amarch/ [Broken]

It mentions, that the which way information is erased AFTER it made click. (at least that's how I understood it) In your gedankenexperiment I now think that the interference is destroyed, that is you don't get any interference although you take away the mirror AFTER it made click (at the center-detectors). However, that's my first thought after having read about these quantum erasure experiments.

I will let you know, if I get any new insight into this.

Regards

Edgardo

P.S. Check out this program about quantum erasure.
http://www.physik.uni-muenchen.de/didaktik/Computer/interfer/interfere.html
Note that you can rotate the polarizers, which I at first didn't get.
 
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  • #15
Talking of erasing the past, in Roland Omnes's Understanding Quantum Theory, he mentions the ideal von Neumann experiment.

A particle is emitted and can go down one of two channels. There is a detector in one channel. If, after a certain time, the particle is detected then you know it went along the channel with the detector and if the particle is not detected then you know it went along the channel without the detector.

So far, so simple. :smile:

Now here comes the interesting part as, if you arrange it so that the two channels curve and later join up into the same direction, then when the particle reaches the point at which the channels join up then the detector result will go from "detected" or "not detected" into a superposition and "don't know" Schrodinger Cat state even if the particle is now a huge distance away from the detector! :bugeye:

It's EPR and Schrodinger's Cat in one simple, fun package. :biggrin:

Decoherence keeps this lunacy from happening in the real world, thankfully. :wink:
 
  • #16
caribou said:
Talking of erasing the past, in Roland Omnes's Understanding Quantum Theory, he mentions the ideal von Neumann experiment.

A particle is emitted and can go down one of two channels. There is a detector in one channel. If, after a certain time, the particle is detected then you know it went along the channel with the detector and if the particle is not detected then you know it went along the channel without the detector.

So far, so simple. :smile:

Now here comes the interesting part as, if you arrange it so that the two channels curve and later join up into the same direction, then when the particle reaches the point at which the channels join up then the detector result will go from "detected" or "not detected" into a superposition and "don't know" Schrodinger Cat state even if the particle is now a huge distance away from the detector! :bugeye:

It's EPR and Schrodinger's Cat in one simple, fun package. :biggrin:

Decoherence keeps this lunacy from happening in the real world, thankfully. :wink:
This doesn't seem correct to me. The detector should go into a superposition of "detection" and "non-detection" even if you don't join the channels.
 
  • #17
some thoughts on the thought experiment

This thought experiment has captured my interest, and I have been thinking about it for some days now. I do not know much about "quantum erasure" and ((S)P)DCs, and so I have only been able to take the setup at face value on the basis of heuristic considerations.

The following "rule" appears to remove all elements of inconsistency:

Interference effects will occur in the signal only if the 'decision' to erase the "which-way" information occurs a spacetime event which lies in the causal past of the signal detection event.

If one would like to argue that this "rule" does not apply, namely that interference effects in the signal exist even when

(i) the 'decision' to erase and signal detection are spacelike separated

or, much more drastically,

(ii) the 'decision' to erase lies in the causal future of signal detection ,

then, in case (i) we can arrange for faster-than-light signaling, and in case (ii) we can arrange (with some abuse of the term 'decision') for an action performed now to 'undo' an event which has already taken place (... whatever that could mean).

I think that this "rule" and the ramifications of its violation for cases (i) and (ii) above apply quite generally to all erasure-type scenarios, and not just to the thought experiment at hand.
 
  • #18
Fredrik said:
This doesn't seem correct to me. The detector should go into a superposition of "detection" and "non-detection" even if you don't join the channels.

I'll quote a couple of relevant parts:

This shows the most problematic aspect of an ideal measurement: the data it yields are not obtained once and for all. Apparently lost interferences can be regenerated later in the measuring device by an action on a distant system (the particle). There is no possibility for considering experimental facts as being firmly established. One may see the result as a particularly vicious consequence of EPR correlations or express it by saying that Schrodinger's cat cannot be dead once and for all, because evidence for his survival can always be retrieved.

Another part, explaining a diagram of the apparatus:

A von Neumann ideal meaurement does not produce a factual datum. The particle shown in Figure 19.1 is brought along direction 3, whatever its initial direction was. When the two paths join together, the state of the measuring device becomes a pure state, with no memory of a previous mixed state where the result of a measurement could be read.

I can write up the full thing if you want. It's just over a page long. I'd just need to learn to do equations on this board and attach a diagram.

I'm still learning quantum theory, so maybe I'm messing up somehow. :smile:
 
  • #19
Eye_in_the_Sky said:
The following "rule" appears to remove all elements of inconsistency:

Interference effects will occur in the signal only if the 'decision' to erase the "which-way" information occurs a spacetime event which lies in the causal past of the signal detection event.

If one would like to argue that this "rule" does not apply, namely that interference effects in the signal exist even when

(i) the 'decision' to erase and signal detection are spacelike separated

or, much more drastically,

(ii) the 'decision' to erase lies in the causal future of signal detection ,

then, in case (i) we can arrange for faster-than-light signaling, and in case (ii) we can arrange (with some abuse of the term 'decision') for an action performed now to 'undo' an event which has already taken place (... whatever that could mean).
I'm glad you joined the discussion Eye_in_the_Sky. I've read some of your previous posts and I think they're all very good. This one is too. You provided me with a piece of the puzzle, and helped me understand a few things better than I did before.

You said "in the causal past of" when most people would have said "at an earlier time than". I like that. The "causal past" is of course the set of spacetime events that are at an earlier time for all observers.

However, I have a feeling that the rule you're suggesting is too simple. It seems to me that the rule implies that the "collapse" of the wavefunction is something that happens with a finite speed. In EPR type experiments, the collapse doesn't behave this way, and I think that's a good reason to expect that it won't in this one. I don't expect a phenomenon like "wavefunction collapse" to behave differently in different experiments. (I'm not sure why. I'm using a lot of intuition here).

I have a few ideas of my own about how to explain this experiment...

When we write down a path integral expression for the amplitude of a particle's propagation from event A to event B, the path integral is a sum over all possible paths through space-time (i.e. all possible world lines). It's not just a sum over all paths through space. That means that we in general must consider paths that correspond to faster-than-light (FTL) and slower-than-light (STL) propagation. The reason we often just talk about paths through space is that the contribution from FTL and STL paths is negligible in comparison with the contribution from paths that look like classical world lines. This is the case e.g. in a standard double-slit experiment.

I suspect that in this experiment, and in others that appear to lead to paradoxes, a careful examination of the path integral expression would reveal that the contribution from FTL paths is large.

The reason I like the FTL paths here is that they seem to provide a "mechanism" (at least on a mathematical level) that let's the "half" photons on the outer paths in the picture "probe" the region in front of them and "determine" if the half-silvered mirror is in place.

When one of the lower detectors is about to click, the "half" photons on the outer path have already "determined" if the half-silvered mirror is in place, so the "half" photons on the inner path will "know" if they're allowed to go to the right detector or not.

If my explanation is correct, we can rule out your case (ii), but definitely not case (i). We could send signals faster than light from the location of the half-silvered mirror at the upper right to a recipient near the lower detectors. This could be a serious problem for causality (or my explanation :smile:), but I don't think it has to be. I will have to think some more about this to be sure

If my explanation is correct, the decision to erase the which-way information can be made "after" the signal detection event. However it can't be made at an arbitrarily late time. The decision event must be either in the causal past of the signal detection event, or spacelike separated from it.

What do you think?
 
  • #20
It's elegant, but wrong.
 
  • #21
Chronos said:
It's elegant, but wrong.
That's certainly possible. The possibility of sending signals between events that are spacelike separated bothers me, and might be a good reason to dismiss this explanation.

If you can say that it's wrong with such certainty, can you also tell me (and everyone else who follows this thread) what the correct solution is?
 
  • #22
Hello Fredrik,

I have taken some time to surf the web and look at various quantum erasure experiments.

In some experiments the topology of the setup requires that the erasure event take place in the causal past of the signal detection event. So, in these cases there is no potential problem.

In other experiments, the setup topology allows for the erasure event to occur at spacelike separation from signal detection. So, here we might suspect the possibility of being able to use the apparatus for FTL signaling. But this is not so. Why? ... Because in these scenarios the interference at the signal can only be inferred when the associated data set is compared with another data set generated at (or after) the eraser. In other words the "nonlocality" is to be found only in the correlations – just like in Bell.

Also, in these experimental setups (where erasure and signal detection are spacelike separated) it is possible, at least in principle, to arrange for the erasure events to not just be spacelike separated from the signal detections but even to lie in their causal future. However, no problem ensues here. (Even though the experimenter at the eraser can receive a message containing the signal detection results before the 'decision' (to erase or not to erase), that data cannot be used to create a paradox (since we still need the other data set ... which does not even exist yet).)


So, now I see that the thought experiment which you have proposed is of a fundamentally different character from any of the typical erasure experiments. Hmm ...

I think that before I attempt to understand better just how this thought experiment can have the purported results, I had better review the entire setup again asking myself, "What did I do wrong? ... when I thought it was right."


P.S. I am not sure I understood your explanation in terms of the path integral. In particular, how is it that case (ii) is ruled out. If your explanation is valid, then why can't the outer "half" photons probe the future all the way up to the causal future of the signal detection event?
 
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  • #23
Eye_in_the_Sky said:
So, now I see that the thought experiment which you have proposed is of a fundamentally different character from any of the typical erasure experiments.
That's why it's so interesting (or just plain wrong).

Eye_in_the_Sky said:
In particular, how is it that case (ii) is ruled out. If your explanation is valid, then why can't the outer "half" photons probe the future all the way up to the causal future of the signal detection event?
I see what you mean. If virtual photons "probe" spacetime, they should be able to probe all of it except the causal past. The only way I see to save my explanation is to postulate that only FTL virtual particles can probe spacetime. With that additional assumption, we can rule out (ii) but not (i).
 
  • #24
I regret my choice of words in the previous post (#23). An additional postulate is the last thing we need. If it makes sense to think of FTL paths (or FTL virtual photons) as probing the surrounding region of spacetime, it must be possible to prove this by evaluating the path integral explicitly (or by using the methods of perturbative quantum field theory).

I still have strong doubts about my own explanation. If it is correct, it is possible to send a message consisting of a single bit over a spacelike interval. It will of course fail 50% of the time, but that doesn't seem to be enough to avoid the usual paradoxes associated with superluminal signals.
 
  • #25
[EDIT: There is some need to edit this post. All such changes will appear in "red" (or some other color).]

[NOTE: A second round of editing now appears in "blue".]

Hello Fredrik.

About that "rule" (which I suggested back in post #19), concerning which you indicated:
Fredrik said:
It seems to me that the rule implies that the "collapse" of the wavefunction is something that happens with a finite speed. In EPR type experiments, the collapse doesn't behave this way, and I think that's a good reason to expect that it won't in this one. I don't expect a phenomenon like "wavefunction collapse" to behave differently in different experiments.
Yes, yes. I agree. When I wrote down that "rule", I had in mind (ever so vaguely) that there could be a way to 'play around' with the idea of 'a decision' and thereby somehow avoid the conclusion which you have pointed to. But that (vague) 'reasoning' of mine turns out to be quite fallacious. Moreover, at that time, I was not at all appreciating the severity of taking such a "rule" seriously.

... And even more delusional was my thought:
I think that this "rule" and the ramifications of its violation ... apply quite generally to all erasure-type scenarios, and not just to the thought experiment at hand.
.......

That being said, we still have this thought experiment to deal with; and if our current understanding of it is correct, we are then – of course – FORCED to take the "rule" (or at least a slightly weaker version of it) SERIOUSLY!

However, as I indicated in my last post, I was not yet ready to do that, opting instead for a more careful reconsideration of the setup.

On the other hand, you yourself have elected to examine this very question, posing it as follows:

How can one 'derive' such a "rule" (or its weaker version) in terms of the mathematical machinery of path-integrals?

In the course of your deliberations you arrived at:
Fredrik said:
The only way I see ... is to postulate ...

... I regret my choice of words ... An additional postulate is the last thing we need.
The "rule" (even its weaker version (which allows for FTL signaling)) is not consistent with conventional Quantum Mechanics. Therefore, the only possible way to 'derive' it – if at all! – would be through the agency of NEW postulates.

For me, such a prospect is more than reason enough to go back and look for some misconception in the original analysis of the thought experiment.

Let us do that now.
________________

The hint of just where our misconception lies is to found in the very words:
... the rule implies that the "collapse" ...
Indeed, in every "quantum-eraser" scenario, the which-way information is "erased" by means of:

(a) the removal of a "measuring instrument" from the setup ,

or

(b) the insertion of a "measuring instrument" into the setup .

[EDIT: My attempt to generalize has failed. In all of the type-(a) scenarios which I have considered, the which-way information is not "erased". Rather, a which-way 'tagging' scheme has been "removed" from the setup.]

[EDIT: In connection with case (b), the prescription that the inserted "element" be of the "measuring-instrument" genre is required only when:

the so-called "erasing" action takes place at the site of signal "counterparts" which exist in an appropriate state of quantum entanglement with the signal.]


By the term "measuring instrument" I mean some entity (or entities) whose 'action' on the quantum system is represented in the formalism by a PROJECTION operator.

Now, take a look at our alleged "eraser". It is nothing but a half-silvered mirror. The 'action' of such an entity is represented by a UNITARY operator. That is to say, in our current scenario, this 'action' is strictly local affecting only the outer "half"-photons while leaving the inner ones completely undisturbed. In other words, by inserting the upper half-silvered mirror into the experimental arrangement, the which-way information is not destroyed, but merely (and this point I think requires further consideration) rendered 'inaccessible'.

[The last sentence in the above paragraph fails to properly convey the required idea. I would like to rewrite that sentence as follows:

In other words, by inserting the upper H-mirror into the experimental arrangement, we eliminate the which-way information for the outer "half"-photons ... but this has no effect on the inner correlated counterparts whatsoever because there is NO COLLAPSE induced by that H-mirror.

As for trying to give an equivalent characterization of this point in terms of such concepts as "destroyed" or "rendered inaccessible", further consideration is required.]

Consequently, it must be the case that the lower detectors always click with 50:50 likelihood, even WITH the upper half-silvered mirror in place.

We therefore do not have a so-called "quantum-eraser" scenario here.
 
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  • #26
Hello Eye_in_the_Sky,

as far as I understood, Fredrik's experiment isn't about quantum-erasure,
but rather the opposite. "Quantum erasure" is the phenomenon, that you can erase the which-way-information, right?

But in Fredrik's gedankenexperiment, we do the opposite, we try to gain which-way-information by taking away the upper half silvered mirror.

I think the question is really what happens in the inner Mach-Zehnder-system, that is, what kind of statistics do we get there?

In my opinion, the interference is destroyed although we take the upper mirror away after it made click at the inner detectors. I know, that seems strange, but I got this impression after reading about the delayed quantum erasure experiments.

What do you think is the statistics at the inner detectors in Fredrik's gedankenexperiment?
 
  • #27
Edgardo said:
... as far as I understood, Fredrik's experiment isn't about quantum-erasure, but rather the opposite ... in Fredrik's gedankenexperiment ... we try to gain which-way-information by taking away the upper half silvered mirror.
First look at Fredrik's setup without the upper half-silvered mirror in place; call this new setup E1. Then put back the upper half-silvered mirror to return to Fredrik's setup; call this setup E2.

Next, consider the following two statements:

S1) In setup E1 the detection ratio for the lower detectors is 50:50 ;

S2) In setup E2 the detection ratio for the lower detectors is 100:0 .

For the moment, let us assume that both of these statements are true. Then, E2 – when compared to E1 – has the character of a "quantum-erasure" experiment, and the upper half-silvered mirror plays the role of the "eraser".
___________
Edgardo said:
What do you think is the statistics at the inner detectors in Fredrik's gedankenexperiment?
50:50. Statement S2 is false.
 
  • #28
Thanks for the answer Eye_in_the_Sky. You've given me a lot to think about. I've been thinking about it a lot since I read it, but I need to think some more...
 
  • #29
a concise summary

Below is a generalization based upon what I have seen and found regarding Quantum Erasure.

Following that are some comments in connection with Fredrik's thought experiment.
_____________

In all of these "quantum-erasure" experiments, the which-way information of the signal is "erased" by means of the insertion of a new element – called the "eraser" – into an existing setup. The insertion-point of that "eraser" can be in one of two places, depending upon the type of setup involved; that is, the "eraser" can be inserted:

1) in the signal beam;

or

2) not in the signal beam, but in some other beam consisting of a signal "counterpart" which exists in an appropriate state of entanglement with the signal.

In a type-2 scenario, the are two important points to take note of:

2a) the "eraser" must be something which 'collapses' the wavefunction;

2b) the statistics at the signal detector(s) are the same whether or not the "eraser" has been inserted.

Regarding 2b, the data set at the signal detector(s) must be compared with another data set generated at (or after) the "eraser" in order to be able to infer that a 'change' of some kind has occurred in the signal on account of the "erasing" action.
_____________

In connection with Fredrik's thought experiment (which was of the type-2 genre), in the original analysis it was found that condition 2b was violated. In that analysis, however, it was implicitly assumed that a half-silvered mirror (serving as the "eraser") is something which 'collapses' the wavefunction ... which, of course, is not true. That is to say, the requirement of 2a was not met, yet implicitly in the original analysis that condition was assumed to be true.

Since 2a does not hold, that setup does not give rise to a "quantum-erasure" scenario.

In more intuitive terms, it seems reasonable to say that, yes, the which-way information of the signal "counterpart" (i.e. the outer "half"-photons) is erased, but, because the "eraser" itself (i.e. the upper half-silvered mirror) does not 'collapse' the wavefunction, that "erasing action" is not (so to speak) 'communicated' to the signal.

From another perspective, one might venture to add that, by inserting the upper half-silvered mirror into the setup, the which-way information of the signal which was present in the entangled signal-"counterpart" has merely been 'thrown away', or in other words ... 'ignored'.
 
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  • #30
I haven't studied all the messages carefully, but had a look at the diagram and can predict that you won't actually ever be able to get the 100% to 0% split you expect. Instead, you should find that you get signs of interference effects showing up if you look at the coincidence curve for the two lower detectors.

You might be able to get your predicted effect if you followed the design of, say:

Zou, X Y, Wang, L J and Mandel, L, “Induced coherence and indistinguishability in optical interference”, Physical Review Letters 67, 318 (1991)

If I remember correctly, they took the output from one DC and fed it into the other. They interpreted their effect in terms of indistinguishable paths, but I think there is a much simpler classical explanation: the fact that a single laser is used as main input to both DCs means that the output phases are linked to those of this laser but, because the frequency is only half, 50% tend to be linked to "even" laser peaks, 50% to "odd" ones. With the DC's independent, the choice of even or odd is independent, but when they are linked this forces them to make the same choice.

There was an article that included the experiment (though not my personal explanation!) in Physics Today, or was it Physics World. See:

Greenberger, Daniel and Anton Zeilinger, “Quantum theory: still crazy after all these years”, Physics World, 33-38, September 1995

or maybe

Greenberger, D M, Horne, M A and Zeilinger, A, “Multiparticle interferometry and the superposition principle”, Physics Today, 22-29, August 1993

Cat
 

1. What is a delayed choice experiment with a paradox?

A delayed choice experiment with a paradox is a thought experiment that challenges our understanding of cause and effect in quantum mechanics. It involves a setup where the outcome of a measurement on a particle can be influenced by a decision made after the particle has already passed through a series of detectors.

2. How does a delayed choice experiment with a paradox work?

In a delayed choice experiment with a paradox, a particle is sent through a series of detectors that can measure its properties. However, the final detector can be set up in two different ways, one that will reveal the particle's path and one that will not. The decision on which setup to use is made after the particle has already passed through the previous detectors, creating a paradox of whether the particle's behavior was determined by the final setup or by its previous interactions.

3. What is the significance of a delayed choice experiment with a paradox?

A delayed choice experiment with a paradox challenges our understanding of causality and the nature of reality in quantum mechanics. It suggests that the act of observation or measurement can have a direct influence on the behavior of particles, even after they have already interacted with other particles.

4. What did I do wrong if my delayed choice experiment with a paradox did not produce the expected results?

There are many factors that can affect the outcome of a delayed choice experiment with a paradox, including the setup of the experiment, the accuracy of the detectors, and the precision of the measurements. It is important to carefully control and monitor these variables to ensure accurate results.

5. How can a delayed choice experiment with a paradox be applied in real-world situations?

While a delayed choice experiment with a paradox is primarily a thought experiment, it has implications for our understanding of quantum mechanics and the nature of reality. It can also be used to develop new technologies, such as quantum computers, that rely on the principles of quantum mechanics.

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