Quantum theory of measurement

[goertzel]

Hi,

I'm writing something that touches on issues of quantum measurement (though the main theme is something different), and I want to be sure I'm not saying anything wrong about this very tricky subject.

So I'm going to pose some thought-experiments to the physics experts on this forum.

The questions I have in mind pertain to the nature of quantum measurement in general, but for sake of concreteness I'm going to pose them in the context of a particular quantum phenomenon, the "quantum eraser."

Specifically, I'll refer to the quantum eraser thought experiment summarized at

http://www.dhushara.com/book/quantcos/qnonloc/eraser.htm

though the same point could be made in regard to other similar (real or gedanken) experiments.The questions I have regard the replacement of the Coincidence Counter (from here on: CC) in the

above experiment with a more complicated apparatus.

What if we replace the CC with one of the following:

1) a carefully sealed, exquisitely well insulated box with a printer inside it. The printer is
hooked up so that it prints, on paper, an exact record of everything that comes into the CC. Then, to "erase" the printed record, the whole box is melted, or annihilated using nuclear explosives, or whatever.

2) the same as 1, except the box has a person (let's call him Fred) inside who reads the printout.

3) a printer that is not encapsulated in a box. There is a person (Fred) standing near the printer who, when he takes each sheet of paper out of the printer, puts it in an envelope and seals the
envelope, but without ever looking at the paper. Then, to erase the printed record, the envelopes are burned, or nuclearly annihilated, or whatever.

4) the same as 3, except that Fred looks at each sheet of paper before putting it in the envelope.
But Fred never communicates anything about what he sees to Bob, who is the guy looking at the screen to see if there's a wavelike or particlelike output pattern on it. The erasure occurs by annihilating both Fred and the envelopes.

5) the same as 4, except that Fred and Bob are not individual people but the two brain-lobes of a person who has experienced a severance of the corpus callosum. Thus Fred cannot communicate
information to Bob even though they are "in the same head." (OK, this one is proposed purely for entertainment value ;)

6) one can redo the experiments with Fred using a pigeon instead of a human, where the printer is replaced by an apparatus that flashes colored rectangles on a touch screen, and the animal is encouraged to pay attention to the screen by being forced to touch the rectangles to get food.

What will the outcome be in these experiments?

Has anyone performed experiments like these? (1, 3 and 6 are the onIes that could be humanely carried out, obviously.)

My own understanding is that whether Fred, a pigeon or a printer is involved in the experiment should be basically irrelevant. That is, I don't think "registration in consciousness" (whatever that means) is the important thing, but rather registration in the sense of "statistical correlation

with some macroscopic system effectively obeying classical probability theory." However, I realize that not everyone agrees with me on this; my reading of Penrose, for instance, is that he would predict a different outcome for 2 versus 1, because he believes that Fred's brain (via unspecified quantum gravity related effects) does something special to collapse the wave function, which the printer does not.

Furthermore, my own understanding is that, so long as

a) the information from the CC is not statistically correlated with the observer (Bob) who is observing the screen, prior to its "erasure," and

b) the erasure is complete, i.e. there is really no way to reconstruct the information after the erasure is carried out then the situation should be just like in the original quantum eraser experiment.

Regarding a, note that statistical correlation is transitive to a significant degree, so that if the information from the CC is correlated with some entity Y (such as Fred, or a printer) that is in turn correlated with Bob, then this means the information from the CC is correlated with Bob.

The main open question I see regarding this "statistical correlation" interpretation is whether it's really possible to adequately annihilate a macroscopic entity like Fred or a printer, and whether it's really possible to adequately isolate two macroscopic entities (like Bob vs. the printer or Bob
vs. the envelope or Bob vs. Fred). One could argue that there will always be subtle correlations between Bob and these other things (since e.g. both Bob and Fred almost surely contain atoms that once belonged to George Washington), and that these subtle correlations mean that in all these
scenarios there actually is some statistical correlation between the screen and the contents of the CC, via transitive correlations such as

CC --> printer --> George Washington --> Bob --> screen

I don't like this argument because it seems to me that these correlations should be too small to make any difference, but I recognize it as an apparent mathematical loophole, according to which my variant scenarios may be considered as different from the original quantum eraser experiment.

Comments?

-- Ben Goertzel

 

[vanesch]

In my opinion, "quantum erasure" experiments are badly named. What I'm going to write is my personal view, and there can be other views of the situation. But in these matters, "things get personal"
In fact, I don't think that it is the "erasure of information" that is "restoring the interference". It is the fact that in order to observe the interference, you have to measure a quantity which is incompatible with the one that is "extracting information" and as such you cannot do both, and if you try to do so, you simply screw up the measurement.
So it is not because you are going to "destroy the paper" or whatever, that suddenly the interference pattern is going to reappear. You screwed up the measurement already. There is only ONE way to "undo" the measurement, and that is not by "destroying the paper" or something more violent, but by carefully applying the inverse unitary evolution that UNDOES (and not simply destroys) the measurement in a nice, reversible way, so that you get back to the situation prior to the "information extraction" and then you can do your "interference measurement" again with success. Of course, as a by product, you don't have the "information" anymore, so it looks as if you "destroyed" it. But I don't think that "destroying the information" in an uncontrolled manner (by nuking it, for instance) is going to RESTORE the state from which you can obtain interference. In fact, it is even IMPOSSIBLE to destroy the information, EXCEPT BY THE INVERSE UNITARY EVOLUTION. The reason is exactly this unitary evolution in QM: there's no way to get rid of it.
The problem with this "inverse evolution" is that once your information has been amplified to macroscopic level (by printing it, looking at it with eyes and a brain etc...) it becomes for all practical purposes IMPOSSIBLE to reverse this. You cannot "unevolve" the brain, the eye, the ink on the paper etc... to go back to the initial state, and that's what you need to do to UNENTANGLE these extra systems with the original terms of the system, in order for the two terms to get summed together again, and to have them interfere.
cheers,
Patrick.

 

[Tez]

The erasing of the information need not be unitary vanesch, in fact generally it isn't.

goertzel, the link you gave was too boring and densely written to read through, but I am familiar with the experiment. Its seems in your gedanken experiment analysis you are not taking into account that the erasure must be of a special form.

Here is a simpler to understand version of quantum erasure: Consider the standard two slit experiment with single photons, and in particular horizontally polarized photons. Now imagine that a polarization rotator is placed just behind one of the slits. The rotator is transparent so the photon goes through it, its just that now its polarization would be vertical if it had "really" come through that slit.

What happens now is the interference pattern disappears. This happens even if the detection screen is not polarization sensitive and just records flashes where any old photon lands. It disappears because the information is in principle available (one could measure the polarization if one wanted, and then determine which slit the photon came through.)

To restore the interference pattern a simple method is to take a piece of polaroid (that sunglasses plastic which only let's one angle of polarization through) and place it at 45 degrees, just before the detection screen. Now any photon that does go through the plastic has had the information about which polarization it possessed erased. Note that both horizontally and vertically polarized photons are equally likely to get through such a piece of polaroid at 45 degrees. Half the photons are lost (absorbed by the plastic) and so you must run the experiment twice as long to get as many detections. (A cute question is: what would you see if the polaroid was oriented at -45 degrees?). A more sophisticated setup "filters" (postselects) the photons into a 45 degree and a -45 degree "bin", and then every photon does contribute - but that is not crucial to the understanding).

The point is that what really happens is the photon's spatial mode and polarization degrees of freedom have become entangled, and a very specific type of measurement (special plastic at a special orientation) was required to destroy the entanglement in a way that still allows for observation of the interference. A simple "burning of paper" is not going to be a destruction of this specific form for any conceivable experiment that springs to mind...

 

[vanesch]

The erasing of the information need not be unitary vanesch, in fact generally it isn't.

Oh, but it is !
I'll show you:
when you write:

The point is that what really happens is the photon's spatial mode and polarization degrees of freedom have become entangled, and a very specific type of measurement (special plastic at a special orientation) was required

This was NOT a measurement! This was a unitary evolution on the overall photon state. The relationship between the state of the photon (the ENTIRE state of the photon !) before and after your piece of plastic is entirely unitary.
EDIT: or could be made so, by using a polarizing beam splitter instead of an absorbing polarizer, which is not essential to the demonstration of the effect.

to destroy the entanglement in a way that still allows for observation of the interference. A simple "burning of paper" is not going to be a destruction of this specific form for any conceivable experiment that springs to mind...

Exactly, because the "burning of the paper" is not the required INVERSE unitary evolution which factors again the two entangled pieces of photon state. Two entangled pieces can only be factorized again by a unitary evolution: the inverse operator that entangled them in the first place.
Specifically, out of your 2-slit system comes the following state, for instance:
|L,y+> + |R,y+>
(L = left slit, R = right slit ; y+ is polarized in the +y axis, the horizontal axis)
The polarization rotator behind the left slit introduces a unitary evolution U1 of the state which maps:
|L,y+> -> |L,z+>
|L,y-> -> |L,z->
|R,X> -> |R,X>
After this unitary evolution U1, our state is now:
|L,z+> + |R,y+>
If we let this interfere we won't see any interference pattern of course, because the in product of a z+ and a y+ state is 0.

However, if we send this state through the 45 degree polarizer (ok this one is not unitary because it absorbs a part, but if we take a polarizing beam splitter, it will be), which introduces UNITARY evolution:
|A,z+> => |A,up,w+> + |A,through,w->
|A,y+> => |A,up,w+> - |A,through,w->
where w+ and w- are the 45 degree directions, and A can stand for R or L, we can work out that the "through" part WILL give interference, because now they have the w- polarization and the inproduct will not vanish of course.
But this is a unitary evolution, and the best proof of it is that you can USE the up part (in a polarizing beam splitter) again and let it mix with the "through" part to RESTORE the z+ or the y+ states.

So no "information" was destroyed in any of these unitary steps. It is only because, in the end, we decided to measure the "through" beam, that we had an interference pattern, and this measurement is incompatible with the z+/y+ measurement that we could have performed by recombining the up and through beams again.


cheers,
Patrick.

 

[CarlB]

Here is a simpler to understand version of quantum erasure: Consider the standard two slit experiment with single photons, and in particular horizontally polarized photons. Now imagine that a polarization rotator is placed just behind one of the slits. The rotator is transparent so the photon goes through it, its just that now its polarization would be vertical if it had "really" come through that slit.

Following Schwinger's line of reasoning in his measurement algebra, consider a "generalized" polarizer. It takes electrons in of some spin orientation, but spits neutrons out with some specific spin orientation. Again interference is eliminated unless you do the same thing to both legs of the two slit experiment. But most people would explain the absence of interference as being due to the fact that distinct particles do not interfere (at least to first order).

The whole argument also applies to the Stern-Gerlach experiment and electron interference. Spin up electrons do not interfere with spin down electrons.

So my preference is to think of the differently polarized photons as not interfering because they are different particles. Only particles that are identical can interfere (to first order).

Of course when you get to weak force stuff it turns out that a better way of splitting the electron into two particles is according to helicity or chirality, but that's no worse than splitting the photon into circularly polarized beams as opposed to horizontally and vertically polarized.

As an aside, a few years ago some physicists solved the problem of defining the photon propagator in position space, something that a few of my old textbooks says is impossible. But to do it, they had to write the photon as circularly polarized. I think that this, along with the weak force structure, indicates that the distinct particles truly are distinguished by helicity.

Carl

 

[vanesch]

Oh, but it is !
I'll show you:
when you write:
This was NOT a measurement! This was a unitary evolution on the overall photon state.

I will even add to this particular case (one-photon quantum erasure experiment) that the entire setup can also be handled by classical optics. As such, interference or not is only dictated by the particular transfer function of the entire optical system, which can be made reversible (for reversible optical components). This is the equivalent of a unitary evolution operator in the quantum picture of the setup.

 

[Tez]

In conventional quantum mechanics the situation I outlined is described by a measurement: With probability 1/2 the photon gets through the polaroid, with probability 1/2 it doesnt. [caveat: Everything can be made unitary if one wanted to go down the road(s) of many worlds, for instance. I have no time for that, though for quite different reasons than the quantum eraser! And not that I'm suggesting you are advocating such either..] The quantum eraser experiments I know of have always involved similar such measurements, (but I haven't read them all), though I admit its a matter of convenience in most cases I'm aware of. For instance, the one I am very familiar with is http://prola.aps.org/abstract/PRL/v75/i17/p3034_1 and that definitely involved measurement (says so in the abstract I just noticed!)

Note: I simply said "need not be unitary", no need to get worked up :)

 

[vanesch]

In conventional quantum mechanics the situation I outlined is described by a measurement: With probability 1/2 the photon gets through the polaroid, with probability 1/2 it doesnt.

With an absorbing polariser, that is correct of course. However, as I pointed out, this is not an essential part of the setup, because, you will agree with me, exactly the same effect can be observed with a polarising beam splitter with a reversible effect. It is the very fact of being reversible that makes the transformation unitary. If you consider it a measurement (meaning that one applies the Born rule IN THAT BASIS ; or which comes down to putting to zero the non-diagonal elements of the density matrix), THEN IT WOULD BE IMPOSSIBLE IN PRINCIPLE TO RESTORE THE ORIGINAL STATES of y+ and z+. And that is manifestly not true as I pointed out: you can make the outgoing beams of the polarizing beamsplitter mix again and restore the original states (exactly because the polarizing beam splitter is a reversible apparatus).

[caveat: Everything can be made unitary if one wanted to go down the road(s) of many worlds, for instance. I have no time for that, though for quite different reasons than the quantum eraser! And not that I'm suggesting you are advocating such either..] The quantum eraser experiments I know of have always involved similar such measurements, (but I haven't read them all), though I admit its a matter of convenience in most cases I'm aware of.

Well, it happens that I AM an advocate of MWI, but I agree with you that this has nothing to do with it here. Orthodox QM considers REVERSIBLE actions never as measurements but always as unitary transformations ; MWI considers ALSO irreversible actions as unitary, while orthodox QM calls this "measurements".

In both views it is simply an ERROR to say that a photon that went through a beamsplitter or a hole underwent a measurement because you can always transform the state back to the original. It is because one wants to enforce too early a classical picture that one makes this error, and the best way to find out is if it is *in principle* possible to restore the original state. As long as this is *in principle* possible, the evolution is to be considered unitary (and reversible).

For instance, the one I am very familiar with is http://prola.aps.org/abstract/PRL/v75/i17/p3034_1 and that definitely involved measurement (says so in the abstract I just noticed!)
Note: I simply said "need not be unitary", no need to get worked up :)

I'm with family and have a bad access to the net, I'll look at it next week.

What "works me up" with many quantum erasure experiments is that one introduces unnecessary magic that confuses the subject, as if information possession somehow makes or prevents the appearance of interference patterns ; while it is mostly just an illustration of the incompatibility of different observables and according to the basis in which one measures the system, one gets one or the other result, and not both.
The problem with the - in my opinion unnecessary - "erasure of information makes fringes appear" explanation (which, as I tried to point out, is only a result of calling certain reversible processes "measurements" and treating them as classical results) is that it automatically calls for black magic of faster-than-light telephones: I erase information HERE, and you see fringes THERE, which is of course totally impossible in QM but which SHOULD be possible if the "information erasure" explanation were correct.

These experiments have nothing special over the position-momentum incompatibility: if I arrange things so that I measure position, I have "erased" the momentum information, and if I change the setup so that I measure momentum I have "erased" the position information ; which should be recognized as a very poor explanation because in the wavefunction there WAS not such a thing as position information or momentum information before measurement. We just applied the Born rule in two different bases.

 

[Tez]

Hey Patrick, sorry I forgot to reply to this.

(Rats - writing long posts is annoying because when I "preview" I lose the spacing!)

I think you're being affected in your views against the standard way people talk about the quantum erasers, "information" and the use of measurement by your interpretational leanings towards MWI. (I'd prefer to avoid the interpretational level of arguments).

So how about a modification which I think better reflects the sort of thing Ben had in mind in the OP: Say instead of using the polarization rotator behind one of the two slits I instead place a single atom in its ground state behind each slit (I guess I really only need one atom behind one slit). I arrange it so that if the photon interacts with the atom then it goes into some metastable excited state. The photon then heads off and impinge on the screen.

Clearly the choice as to whether to erase the information can now be made by the person back at the slits. While it would be possible to bring the photons and the atoms back together again and undo the entangled state, this would be a radically different experiment. I see no harm with the standard explanation: The choice of measurement can either reveal the information as to which slit the photons went through, or not. If the former measurement is done then the person at the slits will not be able to send (classical bits of) information to the person at the screen enabling them to reconstruct the interference pattern. The other choice of measurement will however "erase" that which path information, and (once the appropriate classical information is transmitted to the person at the screen) the interference pattern can then be reconstructed. There is never a possibility of superluminal signalling (not saying this for you, more for Ben if he's still reading!).

With that sort of experiment in mind, I think Ben's original "little friend in the box" question is best answered by saying that the little friend is not performing the right erasing measurements, rather than by invoking the potential for unitarily undoing of the measurement.