Undergrad EPR revisited

[anuttarasammyak]

In an inertial frame of reference (IFR), there are two fixed points, A and B, which share an entangled state
$$ \frac{1}{\sqrt{2}}(|0>_A|1>_B+|1>_A|0>_B) $$
At point A, a measurement is made. The state then collapses to
$$ |a>_A|b>_B, \{a,b\}=\{0,1\} $$

We assume that A has the state $|a>_A$ and B has $|b>_B$ simultaneously, i.e., when their synchronized clocks both read time T

However, in other inertial frames, due to the relativity of simultaneity, the moment when B has $|b>_B$ occurs when B’s clock reads a time before or after T

Therefore, may we conclude that the statement “B has $|b>_B$" is not invariant, but frame-dependent?

 

[PeterDonis]

may we conclude that the statement “B has $|b>_B$" is not invariant, but frame-dependent?

You may conclude that that statement, as you make it, is not even well-defined. You didn't specify a time in any frame at which B has that state.

A true statement would be that the time ordering of spacelike separated events is not invariant, but frame-dependent. But specifying an event requires specifying a place and a time in some frame--an event is a point in spacetime.

You are also sweeping under the rug complications that involve quantum mechanics as well as relativity. For example, suppose we say that the measurement on A happens at a particular event--a particular place and time in some chosen frame. You are assuming that the state of B collapses when the measurement on A is made. But at what event does the collapse of B happen? There is no invariant way to specify that--which should be a clue that the intuitive picture of QM you are using is not, in fact, correct--more precisely, it is not relativistic. In relativistic QM, i.e., quantum field theory, the whole idea of "collapse" is much more problematic, because there are no invariant answers to questions like the one I posed just now.

In short, you unfortunately have chosen a bad example to illustrate the relativity of simultaneity.

 

[anuttarasammyak]

Thank you @PeterDonis. I should be somewhat satisfied to know that studies are going on.

 

[PeterDonis]

I should be somewhat satisfied to know that studies are going on.

I'm not sure what "studies" you're referring to, but it's certainly true that there have been plenty of experiments testing both relativity of simultaneity and the QM predictions for measurements on entangled particles.

 

[cianfa72]

For example, suppose we say that the measurement on A happens at a particular event--a particular place and time in some chosen frame. You are assuming that the state of B collapses when the measurement on A is made. But at what event does the collapse of B happen? There is no invariant way to specify that--which should be a clue that the intuitive picture of QM you are using is not, in fact, correct--more precisely, it is not relativistic.

I was reading this again. As far as I can know, the QM framework provides the answer for the joint probabilities of both Alice and Bob's measurement outcomes.

Therefore, for measurement events spacelike separated, you can think of wavefunction collapsing either way (it first collapses when Alice's measurement takes place or the other way around, that is it collapses first when Bob's measurement takes place). In both cases the joint probabilities are the same.

 

[PeterDonis]

the QM framework provides the answer for the joint probabilities of both Alice and Bob's measurement outcomes.

That's correct.

Therefore, for measurement events spacelike separated, you can think of wavefunction collapsing either way (it first collapses when Alice's measurement takes place or the other way around, that is it collapses first when Bob's measurement takes place).

Not really. See below.

In both cases the joint probabilities are the same.

Yes, but what that means is not that "you can think of the wave function collapsing either way", at least not if you are thinking of the collapse as an actual physical event. (If you're just thinking of it as a mathematical device, sure, go ahead, it doesn't matter.) What the time ordering of the measurements not being invariant means, from the standpoint of relativity, is that you cannot think of the collapse as being a physical event at all, because any physical event would have to be invariant, and there is no invariant that tells you at what event a physical collapse would happen.

 

[cianfa72]

(If you're just thinking of it as a mathematical device, sure, go ahead, it doesn't matter.)

Ok, yes.

What the time ordering of the measurements not being invariant means, from the standpoint of relativity, is that you cannot think of the collapse as being a physical event at all, because any physical event would have to be invariant, and there is no invariant that tells you at what event a physical collapse would happen.

Ah, I see. To me this notion of collapsing wavefunction is really an headache

 

[PeterDonis]

To me this notion of collapsing wavefunction is really an headache

You're certainly not the only one.

 

[cianfa72]

You're certainly not the only one.

Have you find any pill as cure ?

 

[PeterDonis]

Have you find any pill as cure ?

Unfortunately, no. Various patent medicines have been offered in the literature on QM interpretations, but none of them are actual cures.

 

[pervect]

My understanding of quantum mechanics is not good. But I've been under the impression that "ordinary" quantum mechanics is not compatible with special relativity. Specifically, Schrodinger's equation is not compatibility with special relativity, and neither is the idea of instantaneous wave function collapse (instantaneous in what frame?).

The Klein-Gordon and Dirac equations are compatible with special relativity, but my not-good level of understanding is that to really solve them properly, one needs Quantum Field theory, not just "ordinary" quantum mechanics. Issues I have seen referred to are negative probabilities (in the Klein-Gordon equation), and the necessity for particle creation and destruction, with the number of particles no longer being an integer, but an operator that can have non-integer expectation values. Sadly, I don't have a good understanding of these issues, but I do note that the OP seems to assume a fixed number of particles in his problem statement.

My understanding of QFT is not good - I have read Feynman's QED, but the fact that it doesn't allow me to actually make predictions or solve anything leaves me unsatisfied. It's rather reminicisnt of Feynman's own remarks about "trickery".

 

[PeterDonis]

Schrodinger's equation is not compatibility with special relativity, and neither is the idea of instantaneous wave function collapse (instantaneous in what frame?).

That's correct. What you are calling "ordinary" QM is a non-relativistic approximation. For relativistically correct QM, you need quantum field theory, as you say. In QFT, in order to compute probabilities, you need to define the specific spacetime events at which measurements take place, and you compute the probabilities from quantum fields that are relativistically covariant. There is no need to apply any "collapse" postulate to the quantum fields, the way you have to with wave functions in non-relativistic QM, so the collapse issue doesn't arise.

 

[anuttarasammyak]

It seems that this problem cannot be properly understood without mastering relativity. Until then, however, how about the following compromise?

Observers in different inertial frames who were making observations in the vicinity of measurement at A hold a meeting and present their respective viewpoints. It turns out that each of them is making a rational judgment. As a conclusion of the meeting, they adopt the earliest possible reading of B’s clock, T−L/c where L is distanbe between A and B, as the common consensus. This is because B’s behavior corresponding to clock readings between T−L/c and T+L/c is unknown, and measurement at B may occur during this interval. The only choice consistent with this possibility is the earliest time, T−L/c.

[EDIT]
Another possible consensus is the latest reading of B’s clock, T+L/c. At that time, everybody agrees on the state $|b>_B$. Therefore, that state, from then on, is regarded as frame-independent.
I believe that quantum computers make use of this situation. For example, immediately after Alice obtains her state $|a>_A$ by measurement, she sends a command to B at the speed of light: either “Forward your state to Charlie for further processing” or “Discard your state.”

 

[PeterDonis]

It seems that this problem cannot be properly understood without mastering relativity.

You don't need to "master" relativity. You just need to understand that one of the lessons relativity teaches us is that physics is contained in invariants. See below.

Observers in different inertial frames

You mean, observers in different states of motion. Every observer is "in" every frame; but each observer is at rest in only one of them.

present their respective viewpoints

Which are irrelevant since their "viewpoints" as you're using that term don't include any invariants, and all of the actual physics, including all of the actual measurement results, are contained in the invariants.

The only choice

...is that you cannot assign a time to the collapse at all, as I've already said, because there is no invariant that tells you what such a time should be.

 

[cianfa72]

In QFT, in order to compute probabilities, you need to define the specific spacetime events at which measurements take place, and you compute the probabilities from quantum fields that are relativistically covariant. There is no need to apply any "collapse" postulate to the quantum fields, the way you have to with wave functions in non-relativistic QM, so the collapse issue doesn't arise.

So basically there is a way out of the collapse of wavefunction. Which are the issues associated with QFT and relativity?

 

[PeterDonis]

So basically there is a way out of the collapse of wavefunction.

Only in the sense that QFT gives a mechanism for computing probabilities that doesn't have the collapse postulate the way ordinary non-relativistic QM does. QFT doesn't solve the underlying issue of "what is really going on".

Which are the issues associated with QFT and relativity?

I'm not sure what you're asking about here. QFT is consistent with relativity; it's constructed explicitly to be that way.

 

[cianfa72]

Only in the sense that QFT gives a mechanism for computing probabilities that doesn't have the collapse postulate the way ordinary non-relativistic QM does. QFT doesn't solve the underlying issue of "what is really going on".

In the sense that the math underneath QFT does not involve the collapse postulate, I believe.

I'm not sure what you're asking about here. QFT is consistent with relativity; it's constructed explicitly to be that way.

So, there is no inconsistency at all between QFT and SR/GR ?

 

[anuttarasammyak]

Collapse of (entangled) wave functions comes from measurement. If not only the concepts of the collapse but also measurement in general are excluded from the theory, I cannot evaluate how useful it is.

 

[PeterDonis]

there is no inconsistency at all between QFT and SR/GR ?

No. You can do QFT in flat or curved spacetime.

 

[PeterDonis]

If not only the concepts of the collapse but also measurement in general are excluded from the theory

What theory are you talking about?

 

[anuttarasammyak]

Collapse of (entangled) wave functions comes from measurement. If not only the concepts of the collapse but also measurement in general are excluded from the theory, I cannot evaluate how useful it is.

If no theories in the direction I have mentioned have been explored in response to the OP’s question, please excuse my oversight.

 

[PeterDonis]

If no theories in the direction I have mentioned have been explored in response to the OP’s question

I'm still a little confused here. The only theories that have been "explored" at all in this thread are non-relatvistic QM, QFT, and relativity. None of them exclude the concept of measurement.

 

[anuttarasammyak]

I'm still a little confused here. The only theories that have been "explored" at all in this thread are non-relatvistic QM, QFT, and relativity. None of them exclude the concept of measurement.

Thank you so much. Then I should appreciate it if you let me know whether all these theories deal with collapse of entangled wave functions as mentioned in OP or not. May I expect that one of them rescue me ?

 

[bhobba]

Thank you so much. Then I should appreciate it if you let me know whether all these theories deal with collapse of entangled wave functions as mentioned in OP or not. May I expect that one of them rescue me ?

These are deep issues not usually tackled at the undergrad level.

I am currently reading a book that tackles them:
https://www.amazon.com/Fields-Their-Quanta-Quantum-Foundations/dp/303172612X

It explicitly makes an assumption - the fields are real. Effective Field Theory casts some doubt on that assumption. Leaving that aside, it examines some recent evidence, such as particles entangled with the quantum vacuum and takes non-locality seriously (I prefer factorisation, but that's another thread). It's
not a cheap book and not easy reading, but it offers a realistic view of Quantum Mechanics compared with other interpretations, such as Copenhagen. A fascinating view of Quantum Theory, that, within its framework, dispels several myths such as the so-called wave-particle duality. Highly recommended reading.

Thanks
Bill

 

[PeterDonis]

I should appreciate it if you let me know whether all these theories deal with collapse of entangled wave functions as mentioned in OP or not. May I expect that one of them rescue me ?

Rescue you from what? I don't understand what issue you think you need to be "rescued" from. If you mean the scenario you described in your OP, I've already responded to that. What part of my response doesn't satisfy you?

 

[Nugatory]

all these theories deal with collapse of entangled wave functions as mentioned in OP or not

Do remember that collapse is not part of the theory of quantum mechanics as described by the math (which is the only real description of the theory). Collapse is an interpretation, a way that we can think about what the math is telling us if we find it helpful. But we don't have to think in terms of collapse, the only reason we would is that we find it helpful... and it is clear from thread that you don't.
(Collapse interpretations do not work well with relativity, so it not surprising that you're finding the idea unhelpful).

So you can stop trying to work out a mental model of what is "really happening" when an entangled wavefunction collapse, go back to what the theory actually says:

If the quantum state is not factorizeable ("entangled" is a standard but not totally precise way of saying this, and we have a bunch of older threads with examples) then if one measurement is performed we can predict the result of the other measurement, when and if it happens or has happened.
That's all.
Everything beyond that is a story we tell ourselves about what is really happening, and if you find the story confusing you can stop telling it to yourself.

 

[PeterDonis]

collapse is not part of the theory of quantum mechanics as described by the math

More precisely, in non-relativistic QM, there is a collapse postulate, but it is not a postulate about anything that "really happens". It is just a mathematical rule for making predictions about further measurements once you know the result of one measurement. Anything beyond that is interpretation, and you don't need any interpretation to make predictions.

 

[anuttarasammyak]

Thank you all. I will think about it further.