Kerr Pre-Measurement vs Bell's Inequality

[sanman]

I have a question, after having read about a non-destructive spin measurement experiment, which was cited as one of the top science stories of 2006:

http://physicsweb.org/articles/news/10/12/15/1#11

http://optics.org/cws/Articles/View...268?articleId=26434&channel=technology&page=1

So that announcement immediately makes me wonder about Bell's Inequality:

http://en.wikipedia.org/wiki/Bell's_inequality#Description_of_Bell.27s_theorem

They say that you can't use the "spooky action at a distance" (correlation violation) to communicate information, since you can't predict/measure in advance what an entangled particle's state will be.

But the non-destructive measurement experiment shows that you can indeed measure the particle in advance, without significantly disturbing/altering that particle's state (or its entanglement?)

Wouldn't this Kerr rotation measurement method then allow for the pre-screening of entangled pairs, based on measurement in advance of state properties like spin?

Couldn't this then be used to exploit the correlation violation (aka "spooky action at a distance") in such a way as to permit its use for communication?

For instance, using the Alice & Bob example, wouldn't it be possible to use pre-measured entangled electron pairs of known spin state, and use the orientation of the apparatus on one end as a way to modulate an information signal, which would then be detected with the other party through the correlation violation?

To me, it would seem intuitive that the answer is yes. Why shouldn't this be able to work?
Please, someone kindly take the time to give me a reasoned reply, even if my post sounds ignorant.

 

[Hurkyl]

I don't know the details, but it sounds something like this:

In a particular basis, let |xyz> denote the quantum state of the two dots and the photon, respectively.

(I will ignore normalization)

By entangling all three, I have put the system into a state such as:

|000> + |111>.

This can be rewritten as

(|00> + |11>)(|0> + |1>) + (|00> - |11>)(|0> - |1>).

Now, if I measure the state of the photon in the {|0> + |1>, |0> - |1>} basis, that means I've collapsed^* the state of the two dots to be either

|00> + |11>

or

|00> - |11>

(and I know which, based on the result of my measurement of the photon). So, the dots remain entangled, although the precise nature of the entanglement depends on what result I got from measuring the photon.




*: substitute whatever way you want to think about the measurement process here. I chose this because it's verbally simpler

 

[sanman]

Hmm,

I appreciate your response, but I wanted to specifically consider the Bell's Inequality aspect to this, since it demonstrates "spooky action at a distance".

Even though it appears to be FTL, we're told to ignore it, since it can't be harnessed/exploited to communicate anything. But to me, that's just a deflective argument -- obviously if something exists, we shouldn't simply ignore it. We need to recognize it and understand why it's happening.

So then if Kerr microscopy can be used to scan an entangled electron without altering it, then why can't it be used to pre-measure/pre-screen entangled electron pairs based on their state, to use these known states as references when we later apply classical measurement techniques on the separated entangled partners?

If I'm able to pre-measure and pre-select entangled electrons based on their state, then suppose I just keep all the spin-Up partners, while giving my friend all their spin-Down entangled counterparts. So I've got a million spin-Up electrons, while my friend has their million spin-Down entangled counterparts.
We then go to opposite sides of the planet.

I then perform the Bell's Inequality experiment, throwing my pre-known spin-Up electrons at a detector whose orientation angle will be selectively altered so as to encode/modulate a signal or information stream. My friend will be able to detect that signal or information stream by detecting correlation violations at his end, as per Bell's Inequality.

The "spooky action" will have transmitted information at a speed Faster-Than-Light as a correlation violation, which is something that's not supposed to be possible. But it was made possible by the fact that we were allowed to pre-measure our electrons before using them with the classical detector.

Therefore, who's right and who's wrong here? How come Kerr microscopy makes it possible to detect an electron's state without altering it? Isn't this a fundamental violation of known physics? And if Kerr detection is perfectly legitimate and truly able to determine an entangled electron's spin without altering it or its entanglement, then doesn't that throw the whole barrier against FTL communication into the trash bin?

 

[Hurkyl]

We have to understand what is happening in the experiment before we can worry about whatever FTL implications it may or may not have.

 

[Doc Al]

I then perform the Bell's Inequality experiment, throwing my pre-known spin-Up electrons at a detector whose orientation angle will be selectively altered so as to encode/modulate a signal or information stream. My friend will be able to detect that signal or information stream by detecting correlation violations at his end, as per Bell's Inequality.

Maybe I'm missing your point, but correlations between measurements made on entangled pairs are only seen after comparing the data collected at both ends. A change in the detector orientation at one end cannot be detected by examining the measurement data at the other end.

If you are thinking of only sending electrons in a pre-defined state, that's something that must be done at the source--before the particles are sent. That "signal" would be the arrival of the particles themselves--nothing FTL about that.

 

[sanman]

Pre-measurement = Pre-Comparison

Well, someone pointed me to this answer:

http://en.wikipedia.org/wiki/No-communication_theorem

 

[sanman]

Hmm, I still however wonder why Bell's Inequality is even possible? Are there any known speculations on why correlation violation occurs? Is there any conjecture on the underlying mechanism behind it?

 

[sanman]

Hmm, that "No Communication Theorem" seems very arbitrarily asserted, but it still doesn't explain why the results of Bell's Inequality are they way they are.

Here's more proof that non-locality is real:

http://www.aip.org/enews/physnews/1998/split/pnu399-1.htm

Doc Al, clearly communication is occurring between the entangled particles, even though we can't make use of it. Otherwise, how do we explain how do we explain "spooky action at a distance"? Can such action occur without communication?

Whether or not we know something has occurred is different than whether it has indeed occurred. The moon doesn't suddenly disappear just because none of us isn't looking at it right at this moment.

 

[sanman]

I found this explanation to be very helpful:

http://www.mtnmath.com/whatrh/node73.html

Quantum mechanics make two seemingly incompatible assumptions. It assumes conservation laws7.11are absolute and it assumes probabilities are irreducible. In classical physics there are mechanisms that explains how the conservation laws are `enforced'. By claiming probabilistic laws are absolute one precludes the possibility of an enforcing mechanism.

Hmm, Probability as its own compulsion? How can it be? To me, that's completely irrational. That's hyper-reductionism. Ockham's Razor run amok.

Einstein explained this difficulty in the previously cited paper known by the initials of its authors EPR[21]. Einstein and his colleagues concluded that quantum mechanics must be incomplete. For momentum must have an objective reality independent of each observation if momentum is conserved absolutely. Nature is doing a sort of cosmic bookkeeping to make sure momentum is never created or destroyed and there needs to be a mechanism not part of any existing theory that implements this accounting procedure. The principle on which Einstein's argument is based is uniformly true in classical physics. It was argued that this principle does not apply to quantum mechanics. The debate is closed for most physicists and decided against Einstein. I suspect Einstein will ultimately be proved correct.

How could one devise further experiments to probe the nature of the accounting mechanism that enforces these probabilities?

Bell's Inequality is just one kind of experiment that reveals the nature of entanglement. What else could one devise that would be usefully revealing, but hasn't been tried yet?

 

[drphysica]

Well there are still a lot of question to be asked because no one really and I mean really knows what a hack is going on in Quantum world. Let's take Bell's theorem which states that no theory of hidden variables can predict results of QM. Here arises a question: QM or locality? But QM has never contradicted a single experiment that was conducted on it (true). So what about the locality? Well here it goes the non-locality is apparent in the theory and yet QFT still has an applications of locality. No-communication theory proves that statistically, Bob cannot tell the difference between what Alice did and a random measurement (or whether she did anything at all) and yet we observe the non-locality in quantum entanglement but doesn't mean that we can transfer the information through space-time by effecting causality. And yet there is an experiment that suggests that the information can be transferred faster then speed of light it was done in Germany and this experiment is the Quantum Channelling of Information faster then speed of light (I can't remember who did the experiment or where to find it on the net. Sorry). You see QM is the theory that predicts the probabilities of particle states and has not certain predictions like we would like. Einstein said that when particle is measured it already carried determined properties (like spin) but we must remember that we only know them when we measure it. QM predicts that we would know a chance for the particle to have spin up or down. We mast be careful when we say things like the moon disappear when we not looking because moon is a celestial object and cannot be treated Q.Mechanically because we could always calculate the position of the moon on the orbit and then observe it to be there with high accuracy. QM'caly we cannot predict particle to be there not here for certain before observations. Ok my conclusion is that QM is a theory, which yet to be explained. I can go into a lot of stuff about it (like parallel universes and so on) just don't have the time. All I know is that it is very arrogant of us to say that we know what da hack is going on (coz we don’t). Indeed we have come a long way since Schrodinger and Einstein but yet the journey is not over. I hope I was helpful.

 

[sanman]

Thank you very much, I really liked your reply.

Alright, fair enough, we cannot treat macro objects like the Moon (or even Schrodinger's Cat) as if they were small particles with quantum behavior.

I had once read about some field of study called Stochastic Electrodynamics ( a derivative of QED) which considers the quantum probability distribution as a form of Brownian Motion, sort of like how a speck of dust floating in a drop of water under a microscope will constantly jiggle, due to the invisible water molecules battering it. Brownian Motion then provides a sort of causal mechanism to explain things like Heisenberg's Uncertainty, DeBroglie Wavelength, and electronic orbitals. The Quantum Vacuum hypothesis says that all of space is filled with virtual particles popping in and out of existence, buffeting real particles and causing their quantum behavior.

I was thinking that the quantum behavior could then be compositely broken down into 1) stochastic probabilistic behavior imparted by the Quantum Vacuum and its Brownian influence, and 2) the particle itself then having intrinsically non-probabilistic behavior.

So that would mean that all quantum probabilistic behavior is due to the surrounding vacuum, and not intrinsic to the particle itself. (ie. Quantum probability is extrinsic)

I was then thinking about how we could test if this hypothesis were true using Bell's Inequality. If we studied various particles that had different strengths of interaction with the vacuum and different strengths of interaction with some entangled partner, then it might help us learn more, because that would allow us to gauge how random the collapse of a wavefunction is. The intrinsic properties of the particle would then become more prominent, with the quantum probabilistic behavior receding somewhat. (ie. more "signal-to-noise" ratio)

The ultra-bright photon experiment mentioned above seemed to indicate that, since photons are less impeded by the vacuum, especially when they are high-energy.
Similarly, what if we tried to study entanglement of a slow but massive system, like a super-heavy superactinide nucleus, or even a buckyball fullerene:

http://physicsweb.org/articles/world/18/3/5

If a buckyball could show quantum behavior DeBroglie wavelength, then could it be entangled? Could one do a Bell's Inequality experiment with it?

 

[sanman]

If non-locality is "spooky action at a distance" and also faster than C, and yet no information is transmitted, then can one have action without information transmission?

Obviously Bell's Inequality is real, so the faster-than-C spooky action is real. But we also know that we can't transmit information with it.

So that means you can have action-faster-than-C without having communication-faster-than-C.

What does it mean to have action without communication? Does that mean String Theory? (ie. the distant entangled partners are actually a single long object spanning the distance between their apparent separate locations, and thus inherently correlated without requiring communication?)

Is entanglement thus the origin of and impetus for String Theory?

 

[CarlB]

If non-locality is "spooky action at a distance" and also faster than C, and yet no information is transmitted, then can one have action without information transmission?

In quantum mechanics, one only ends up with the embarassing (to Einstein's relativity) faster than light spooky action at a distance when one mixes wave and particle in the same experiment.

The waves alone cannot have spooky action at a distance because the waves are local and do not propagate faster than light (at least on the mass shell). And to get particles to mix it up with waves (and so get the faster than light stuff), you have to use the probability postulate that says that actual measured particle positions are proportional to the absolute square of the wave function.

To explain why it is that you can't send useful information faster than light requires a lot more explaining than can possibly fit in one of these threads. A beginning student has two choices. He can resign oneself to not understanding it. Or he can resign himself to spending many years of hard work learning quantum mechanics, and then find out that the experts didn't understand it either.

At heart, the quantum theory is an abstract or "formal" solution to the problem. It doesn't tell you what things are, it just spits out numbers that show how things measure in practice. Since QM is abstract, it can be put into many different formalisms, and yet we cannot know which gives us better insight into the nature of reality. They're all just abstractions.

My problem with the way the paradox is discussed is that the faster than light influence problem requires two incompatible descriptions of the events that mark the beginning and end of the purported faster than light influence. When the correlated quantum objects are emitted from the source they are treated as waves, but then, when we notice the correlations, we are treating them as particles. These are incompatible representations of a quantum object, and since they are incompatible, we cannot use their purported (x,y,z,t) event coordinates in spacetime to compute a velocity of propagation. We're starting the race with apples and using "apple time", but we're ending the race with oranges and using "orange time". These are incompatible event types and we cannot specify a velocity with them.

Another way of saying this is that particles propagate, and we can measure their speed of propagation. That speed is always less than or equal to c (at least for the usual particles in the standard model). And waves propagate. We know the wave theory well, and it propagates at speeds less than or equal to c (barring various unphysical or unlikely cases).

Both these ways of looking at the problem show that quantum objects travel slower than c, either as waves or particles. It is only when we allow the description of the experiment to mix wave and particle that we end up with an alleged faster than light influence.

For example, if we looked at the problem purely from a particle point of view, we would have to measure the particles when they were emitted. This would eliminate the entanglement and there would be no influence moving faster than light. And if we looked at the problem purely from a wave point of view, there would be no probabilities and no influence moving faster than light.

It is only when we demand that the quantum objects begin their race as waves, and end their race as particles that we can define a race where a quantum object has influences that exceed c in speed.

Carl

 

[sanman]

Thanks for your informative reply, Carl.

But let me just put myself in the shoes of the humble Observer.

Alice and Bob have their watches synchronized. Alice and Bob are each given their respective entangled particles, and then travel to opposite sides of the solar system.

At T=0, Alice let's the first particle hit the detector, and keeps releasing another entangled particle every second. Bob is likewise doing the same thing on his end.
At some later time, let's say t=10, Alice changes the detector's orientation, which is supposed to result in a correlation violation once Alice and Bob later get together to compare notes.

So after Alice and Bob reunite to compare notes, at what timestamp will we see the correlation violation first appear? If the correlation violation appears at exactly T=10, then we know the "spooky action" was instantaneous. If there was some delay in Bob's recorded timestamp for correlation violation, then we know that there is a propagation time delay for the "spooky action" to reach where Bob is.

Can we not outfit a space probe with an bunch of entangled particles and an atomic clock synchronized to an earth-based one, and then have it periodically conduct the Bell's Inequality measurement, and also radio back the results each time? (I don't care how long the radio signal takes to get back, I just want to know what measurements and associated clock-timestamps it reports back. Will they identically match the results on Earth?)

 

[drphysica]

I might want to agree with you on that QM probabilistic behaviour is due to surrounding vacuum. The analogy most quantum physicists use is an ocean, that particles in space are like waves in ocean constantly dynamic and waves can appear anywhere with the same properties. So it is possible but not yet observable. The whole observing thing obviously some how alters the quantum space. It feels almost like by testing, through which slit does a wave goes we tap into cosmic hard drive and alter the natural behaviour of waves to get particle behaviour or collapse of wave-function. Some how the law of causality being altered and that produces the collapse of the wave function. Personally I don't think that there is any information is transferred between two entangled particles and it is not because of locality or no-communication theory, but rather of some thing else. I believe that the whole universe is interconnected and that any particles and objects in the universe (including us) are connected. Let's say you are standing on the beach with closed eyes and let's assume that we have no idea of what the waves look like, so when the wave arrives at our feet we feel it and record it that that wave felt in certain way. Now if we move 10 meters to the left or right we again repeat the experiment and observe similar wave. Now the point I'm making with close eyes we have no idea what the true nature of a wave is. Now consider two observers (eyes closed) standing 10 meters apart and observing the same wave, both observers fell the same wave (or measure the same properties) without any information being transferred between the two. Now it is apparent to me and you that we know the nature of the wave and we know it is stretches out for several hundred meters and that both observers experience the same wave and after traveling 10 meters one can tell the other what the nature of that wave is or one can't tell the other the information about the wave its self. What I'm trying to say is that both observers would ask each other how the hack did you know what my wave (particle) felt like since the information cannot travel faster then some limit. The answer is that the both observers will never understand or picture the nature of the particles they observe until they open they eyes and see the whole nature of the universe in font of them. Consider another scenario if one observer (eyes closed) that could simultaneously exist across the whole shore he/she could in fact understand the true nature of incoming wave thus become the wave its self, which actually impossible. Similarly with us we cannot become the wave it self but may be we can open our eyes to the reality. So my reality is that all matter and energy in the universe is just like a magic blanket where everything seems different and yet the same. String theory actually seems like another step closer to understand the reality. I like string theory but it is only at it infancy and it already make cense.

 

[CarlB]

Will they identically match the results on Earth?)

Current theory says yes, the distance between Alice and Bob doesn't change the correlation. I don't believe it, but that doesn't mean we can measure the effect.

All experiments that we have ever made on correlated particles were made on particles that were on the Earth's surface, and therefore separated by a distance that is an almost infinitesimal fraction of the natural length, i.e. the size of the universe. What I'm saying here is that the extrapolation to all possible distances is ridiculously unwarranted; there could be some intervening "new" physics. My guess is that the new physics will be a fading away of the correlation, but the only natural distance scale that immediately jumps to mind would be the universe itself. My intuition says that a distance of a light-second should be enough, but my intuition is an idiot and has a horrible track record -- the only natural distance I can come up with is the size of the universe.

Carl

 

[Doc Al]

Hmm, that "No Communication Theorem" seems very arbitrarily asserted, but it still doesn't explain why the results of Bell's Inequality are they way they are.

That theorem is not arbitrary, but the wiki article overstates its conclusion. One the one hand, it says:

Theorem: In a Bell test, the statistics of Bob's measurements are unaffected by anything Alice does locally.

I agree with that, which is basically what I stated in my first post. But the article starts off by stating:

In these experiments, the no-communication theorem shows that failure of local realism does not lead to what could be referred to as "spooky communication at a distance".

I do not believe such statements are justified by the so-called "no-communication" theorem.

Here's more proof that non-locality is real:

http://www.aip.org/enews/physnews/1998/split/pnu399-1.htm

Doc Al, clearly communication is occurring between the entangled particles, even though we can't make use of it. Otherwise, how do we explain how do we explain "spooky action at a distance"? Can such action occur without communication?

Note that you have changed the subject from your first post, where you were proposing to somehow use entangled pairs to send signals faster than light. That is prohibited by the "no-communication" theorem. But the theorem says nothing about whether non-local connections exist between the entangled pairs. On that topic, I will say that (IMO) I've seen strong arguments made that the results of experiments, coupled with Bell's theorem and the EPR argument, imply that non-locality is very real fact of nature (not just an artifact of a particular theory). (At least if you define locality as Bell did in his theorem.)

 

[sanman]

Thank you DrPhysica, CarlB and Doc Al for your great responses. You've really cleared up a lot for me.

DrPhysica, the problem I have with your analogy of 2 people feeling the same wave on the beach, is that Alice is able to initiate the measurement change that is experienced on Bob's end as the correlation violation. If you and I are on the same beach, I cannot cause a wave to arrive from the ocean simultaneously at your location and mine together, so that we both experience that same thing at the same time.

Carl, I totally agree with you that entanglement experiments have to be done over long distances, so that we can test the strength of non-locality, and ultimately figure out the nature of what is causing the non-local connection.

Doc Al, yes, my curiosity then leans towards wanting to know the underlying mechanism by which non-locality occurs. As you say, Einstein's Relativity and No Communication Theorem hold specifically for classical communication across space, but don't put up any barriers against some other type of communication.

2 possibilities immediately come to mind for me:
1- Entangled particles A and B are somehow "fused" as one object (call it StringAB?) and therefore communication between A and B is really just communication within that object (intra-object communication?) and not communication across space

2- There is some mini-wormhole thing happening, whereby some separate conduit of communication is opened between A and B directly, that bypasses regular space, and yet does not allow information to be transferred. Entanglement of A and B puts them both in their own private universe, where they are immediately adjacent to each other, without any spacetime gap.
Entanglement is then the ability to immediately create a mini-universe consisting of only the entangled objects, without any other spacetime or anything else.

I still think that spooky action-without-communication needs to be better defined, regardless of how fast or slow it is occurring. Does correlation violation really count as "change"?
Are there any other examples of action-without-communication outside of entangled systems? Or is action-without-communication purely a feature of entanglement?

 

[heusdens]

I think that the interpretation of this Bell Inequality experiment is simply wrong.
There isn't non-locality or instantanious communication over large distance.

Suppose the experiment is creation of one particle-antiparticle pair (for the sake of describing, let's make it a electron/positron pair). One particle goes one way, the other the other way.

At a very large distance we detect one particle and identify it as an electron.
Immediately - without even observing - we know the other particle is a positron.

Now QM claims that until we observe one particle, it's in a superposition of two states: an electron and a positron. When we observe the particle, the wave collapses, and we know which state it is in. And by mystery the other particle then also collapses it's state into the opposite particle.

But this description just obfuscates the facts. We can however interpret this experiment in a different way. Instead of saying that each particle is in a superposition of two states (electron and positron) we can also claim that we have a definite electron and a definite positron. Just what we can't find out until we observe it is which way it goes (left or right).
So, actually we have an electron in a superposition of two states (going left and going right) and a positrion also in a superposition of two states (going right and going left). Just that the conservation laws dictate that if one particle goes left, the other must go right.

So, if we decide to inspect the particle going left and find out it is an electron, we know the electron has state 'going left' and for sure we know that the positron has state 'going right'.

Nothing mysterious, and no non-locality or mysterious instantanious communication.

 

[CarlB]

I think that the interpretation of this Bell Inequality experiment is simply wrong.

Your analysis of the electron / positron case is correct, but that's not the Bell inequality experiment which has to do with the far more complicated situation of spin. If the interpretation were that obvious, it wouldn't be discussed for so long.

Electron versus positron only has one possible measurement, charge. Spin has an infinite number of directions it can be measured with respect to, any of the unit vectors in 3 dimensions. This is a lot more complicated.

 

[heusdens]

Your analysis of the electron / positron case is correct, but that's not the Bell inequality experiment which has to do with the far more complicated situation of spin. If the interpretation were that obvious, it wouldn't be discussed for so long.

Electron versus positron only has one possible measurement, charge. Spin has an infinite number of directions it can be measured with respect to, any of the unit vectors in 3 dimensions. This is a lot more complicated.

I will look up that experiment details, but I suspect the same kind of reasoning can be applied.

First of all, as far as I understand the experiment, the descriptions is overly complex, and obfuscates things.

Firstly: Instead of the mentioned two angels which Alice and Bob can switch to, I think there is really just one angle of importance, which is the relative angle Alice-Bob' detection apparatus is set up for each individual measurement.
So in fact we could think of the system as having just one angle of measurement, instead of two.

(note: there is also of course the angle related to the source as well, but changing that angle does not change anything, since the distribution is random, so changing the angle of the source does not alter the experiment, why we can neglect that).

Secondly: is the illogical score systems, which count opposite (related) measurements as 0 (= +1 + -1) AS WELL as unrelated measurements, which also score 0.
So what is that score? It for sure is not the measure of relatedness, since scores of individual measurements of for example: +1, -1, -1, +1 measure 0 and also individual measurements of : 0, 0, 0, 0. So a zero scrore can either mean a VERY related score on individual measurements, or NO related score on individual measurements.

From this it sounds already logic to assume that the highest score would occur exactly in between the measurements of lowest scores, which are the angels of 45 degrees. Adding 180 degrees in this set up, does not change the scores, just that the individual measurements have opposite signs, but since +1 + -1 cancel each other, this does not affect the score.

I think therefore this is rather simple conclusion.

 

[Hurkyl]

Your analysis of the electron / positron case is correct, but that's not the Bell inequality experiment which has to do with the far more complicated situation of spin. If the interpretation were that obvious, it wouldn't be discussed for so long.

Electron versus positron only has one possible measurement, charge. Spin has an infinite number of directions it can be measured with respect to, any of the unit vectors in 3 dimensions. This is a lot more complicated.

That's not quite accurate. The state-space of the electron/positron is the Bloch sphere, just like the spin of the electron. In principle, you could measure it in the

|electron> + |positron>, |electron> - |positron>

basis, or any number of other bases. You could perform the exact same experiment as you would with spin.

Of course, I imagine that the fact the different states have different charges would mean it's difficult to keep the system from decohering through electromagnetic interaction with the environment. (Hopefully someone can correct me if I'm wrong)

But this description just obfuscates the facts. We can however interpret this experiment in a different way. Instead of saying that each particle is in a superposition of two states (electron and positron) we can also claim that we have a definite electron and a definite positron. Just what we can't find out until we observe it is which way it goes (left or right).

If we have an actual superposition of |electron> and |positron>, then that is exactly what QM says we can't claim. What you describe is not a quantum state, but instead a "statistical mixture" of quantum states.

As I said to CarlB, instead of measuring if it's an electron or a positron, I can measure to see if it's an

|electron> + |positron>

or a

|electron> - |positron>.

And I could do the same in a myriad of other bases. With the statistical mixture you describe, all of these measurements will be 50%-50% one way or the other. But if it's a true quantum state, if I measure in the right basis, I can get a 100%-0% split of the probabilities.

 

[CarlB]

That's not quite accurate. The state-space of the electron/positron is the Bloch sphere, just like the spin of the electron. In principle, you could measure it in the

|electron> + |positron>, |electron> - |positron>

basis, or any number of other bases. You could perform the exact same experiment as you would with spin.

The problem is that charge is covered by a superselection rule. That is, in quantum field theory it is known that measurements are subject to a "superselection sector" rule and the above mixture violates that requirement. Can you describe the experimental apparatus that would make the above detection?

The only way I can think of doing such a measurement all would involve bringing the two beams back together. So the experiment as a whole would still be that of an electron and positron, and the off axis basis would only be implied as an intermediate state (about which quantum mechanics is known to be a bit iffy).

As usual, the best introduction on the web consists of some poor guy's dissertation. See the chapter "The Algebraic Theory of Superselection Sectors", which starts on page 6:
http://arxiv.org/abs/math/9809035

The first paragraph of the chapter pretty much states the situation the way I see it:

The theory of superselection sectors is an important and particularly successful branch of local quantum field theoryd. It was initiated by the observation of Wick, Wightman and Wigner in 1952 that the validity of the superposition principle in quantum physics is limited by what they called superselection rules [WWW52]. For instance, there is no interference between a single–electron state $$\psi_-$$ and a single–positron state $$\psi_+$$. In the state $$\psi = \alpha_+\psi_++\alpha_-\psi_-,$$ the relative phase between the $$\psi_\pm$$–components cannot be measured, but can be arbitrarily changed by applying global gauge transformations $$\psi \to \psi' = \alpha_+e^{i\lambda}\psi_++\alpha_-e^{-i\lambda}\psi_-$$. Such $$\psi$$ is not a coherent superposition, but a mixture of the pure states $$\psi_\pm$$, with weights $$|\alpha_\pm|^2$$. Accordingly, matrix elements of physical observables between $$\psi_+$$ and $$\psi_-$$ must vanish, observables are gauge invariant, and the physical Hilbert space splits up into invariant “coherent” subspaces, each carrying a definite value of the electric charge. The unobservability of relative phases in such situations led Wick, Wightman and Wigner to the conclusion that the parities of elementary particles with different charges cannot be compared.

Carl

 

[Hurkyl]

Ah! This is interesting to me. Clearly the operator denoting the experiment exists, just apparently we can't perform the experiment. I must investigate!

 

[CarlB]

I would really appreciate it if you report back what you find. This whole thing about "superselection sector" was never taught to me in grad school and I passed the prelim exams for the PhD. I only found out about it in the last few years, from reading stuff on the foundations of QM.

Somewhere I think there is a statement by Feynman to the effect that the concept that "every Hermitian operator corresponds to a measurement" is not true, and this might have something to do with superselection. I did a search but didn't find it.

But it kind of lines up not with what we (i.e. U. Cal., Irvine, 1982) were taught, but more with what we were taught to do. By that I mean that no one ever discussed a state that was partially positron and partially electron, but they also never discussed why you couldn't do it, nor did they ever assign us homework that would need it.

Maybe the place to search for stuff is on the subject of isospin.

 

[Hurkyl]

I haven't had a chance to go looking, but I have my suspicions. One thing I found interesting is that if I take the C*-algebra generated by two operators:

I and Z

where I is the identity operator and Z satisfies the relation ZZ = I, then IIRC the corresponding state space "is" (I think) the real interval [-1, 1], rather than something like the Bloch sphere you get from a qubit.


On the other hand, one of the nifty general principles in algebraic treatments of geometry is that if you take the quotient space of your geometric object (say... mod out by gauge transformations), that corresponds to taking a subalgebra of the corresponding algebraic object. (say... gauge-invariant functions)


The first thing I describe sounds like it resembles the result you quote. The second thing I describe sounds like it resembles the explanation you quoted. And, happily, the two are consistent with each other:

The algebra generated by 1, Z is a subalgebra of the one generated by 1, X, Y, Z, where we can interpret X, Y, and Z as measuring the spin of a qubit. When we restrict ourself to the algebra <1, Z>, that does the same thing (I hope!) as projecting the Bloch sphere onto the Z-axis, giving us the real interval [-1, 1].


That's why I'm excited -- what you've introduced sounds like a new aspect of QM I can absorb with the knowledge I already have!

 

[CarlB]

It's interesting to speculate on the superselection sectors. The worse part of the superselection rules is that they apply to the SU(2) weak isospin part of the standard model, so you want the usual spin SU(2) operators to act analogously to the SU(2) weak operators (why should nature invent two different ways of making an SU(2) symmetry?).

This all subtly gets back to the question of "superposition and kets" which got discussed here a while back (I don't recall you commenting on it):
https://www.physicsforums.com/showthread.php?t=124904
The crux of the issue is that superselection rules are rules about superposition.

If you read between the lines on the above you might realize that I don't believe in the same universal kind of linear superposition that everyone else does. That is, I think that it is a convenience of the mathematics rather than an attribute of reality correctly exhibited in the physics. For example, twice a ket is a ket that represents the same physical situation, not, as would be true for a truly linear theory, a new physical situation with twice the field strengths.

My view is that different spin particles are truly different particles. When we change the spin of a particle we do it by bringing in a gauge boson which annihilates the initial spin particle and replaces it with a particle of the same sort and changed spin, by creating one with the final spin. In other words, it's a gauge field sandwiched between fermion creation and annihilation operators.

So the real problem is, why is it that gauge bosons exist that change spin, but do not exist that change weak isospin, except as +/-1. My suspicion on this is that weak isospin carries the same orientation information as the particle and so it does change. The information, however, is already contained in the spin, so putting into the weak isospin would be redundant. And the fundamental particles are chiral, that is, they are either left or right handed and massless, and consequently they always take quantum numbers of +/- 1, so it's natural for this sort of thing.

In the usual way of doing physics we don't notice this because we break the particles into left and right handed portions before we deal with them in the standard model anyway. And then the left and right handed portions automatically carry the usual (super selected) weak isospin quantum numbers, +/- 1. There's only one orientable spin because there is only one orientation, and we conveniently put all the orientation information into just one of the quantum numbers.

This gets back to why it is that the left handed chiral fermions are the ones in the doublets while the right handed are in the weak isospin singlets. More accurately, the chiral fermions before they're rotated by the Weinberg angle are pure doublets and singlets. And therefore the un rotated (non chiral) fermions carry their weak isospin quantum numbers exactly parallel or antiparallel to their spin orientation. Under that circumstance, it is very natural for weak isospin to be an oriented quantum number, and a superselection rule is natural.

Uh, I should mention that I suspect that all this speculation is in direct violation to the Coleman Mandula theorem:
http://en.wikipedia.org/wiki/Coleman-Mandula_theorem