Are you distinguishing between a "mixed state" and an "ensemble" and a "mixture of pure states"? Because AFAIK these are all the same thing. If you think they are different, what distinguishes them?
If you agree that they are all the same, then what you are saying is exactly the same as what I am saying, except for the "no entanglement" part. Whether or not you have entanglement or a mixed state depends on what point of view you choose to take. If you have an EPR state, then the joint system is in a pure state, but each individual particle is in a mixed state when viewed as an isolated system. That is what a mixed state is. A mixed state is nothing more or less than the state of a proper subset of an entangled system.
A mixed state can be produced as a subset of a entangled system. It can also be produced as a mixture of pure independent states. They need not correspond to the same physical situation: you can prepare the same mixture state in different ways. But these ensembles can not be distinguished later (unless you “remember” the lost information about the precise state for each element!).
For example, you can get unpolarized light as a mixture of left- and right-polarized light, or as a mixture of horizontal- and vertical-polarized light, or getting one photon from entangled pairs (and discarding the other). I’m sure that you can find a way to describe that as an entanglement with something else, but there is no need to do so.
In particular, when Alice measures a photon in an entangled pair the entanglement is broken. Bob’s photon is no longer entangled with Alice’s photon. Hopefully you will agree that Alice’s photon is now in a pure state. Again, you may want to make things much more difficult but it’s not required to prevent FTL signaling.
> I’m sure that you can find a way to describe that as an entanglement with something else, but there is no need to do so.
That depends on your goals. If all you care about is making correct predictions of experimental results, then yes, one point of view is as good as another.
But if you care about actually understanding what is going on then explaining measurement in terms of entanglement is progress.
> when Alice measures a photon in an entangled pair the entanglement is broken
No. The entanglement remains, and is incorporated into a (much) larger system of mutually entangled particles, including Alice herself. As part of this process, decoherence happens, which makes the original entanglement impossible to detect as a practical matter. But it doesn't go away.
> Hopefully you will agree that Alice’s photon is now in a pure state.
No, I will not agree. When Alice measures the photon, the photon must be absorbed by some detector, and so can longer be said to exist at all in any meaningful sense.
Fine, but this approach leads nowhere. Before you said that “Before teleportation C is in a pure state”. But that’s surely impossible, because wherever C is coming from it’s entangled with a gazillion things already... If the original state C can be “pure”, then the final state B can be “pure” as well (and identical to C).
I think you have a problem with the probabilistic nature of QM, if you only can accept a statistical mixture as being part of an entangled pure system. FTL signaling is just a excuse (and you don’t explain how standard QM would allow for it).
It has led to a great deal of mental clarity for me. YMMV.
> Before you said that “Before teleportation C is in a pure state”. But thats surely impossible, because wherever C is coming from it’s entangled with a gazillion things already...
That's right, and indeed figuring out how a laser actually works is quite challenging.
> If the original state C can be “pure”, then the final state B can be “pure” as well (and identical to C).
Only if you are willing to explain how B was transformed from its initial state into its final state, and more to the point, when this transition happened.
> I think you have a problem with the probabilistic nature of QM
Huh? What have I said that leads you to believe that? It's not true.
> if you only can accept a statistical mixture as being part of an entangled pure system
That has more to do with philosophy than physics. Does an apple really fall because gravity is pulling down on it? No, but it's a good enough approximation to the truth that it's usually not worth quibbling over.
Alice knows which one of the four is the pure state of B after the measurement and tells Bob how to rotate the state (if needed) to recover alpha|0> + beta|1>
So, again, when Alice does the measurement the state becomes one of the four listed states and when Bob rotates the state (if needed) B is the same state that C was originally. Maybe you agree that this is making correct predictions of experimental results, maybe not. I still don't know when and how does B change along the procedure according to your view.
> No, but it's a good enough approximation to the truth that it's usually not worth quibbling over.
Talking about "pure" states after a measurement is a good enough approximation to the truth, but you find it worth quibbling over :-)
> The problem with that description is that the word "now" is not well defined when Alice and Bob are far apart.
Do you have an alternative description to work around this "problem"?
QM is a non-relativistic theory. In this case, the measurement on C/A is strictly before the rotation on B so there is not even the shadow of a problem. Relativistic extensions to QM are Lorentz-invariant and for space-separated observables there is a restriction for operators to be commuting (so the outcome is consistent with either of the observations being "first").
> the measurement on C/A is strictly before the rotation on B so there is not even the shadow of a problem
Yes, obviously. That is not what I'm talking about.
Consider this variation on the theme: Alice and Bob share N>>1 entangled A-B pairs. Alice prepares N additional C particles, all in a known pure state P, and runs them all through the teleportation protocol. But instead of sending Bob rotation instructions, she sends him only the serial numbers of the particles that do not require Bob to perform a rotation. Bob keeps the particles corresponding to those serial numbers, and destroys the rest. I presume you will agree that Bob now has a collection of approximately N/4 particles in a pure state P. (I do not agree with this, BTW, but I accept it here for the sake of argument.)
Now imagine that Bob simply selects N/4 particles at random without any information from Alice. He has no idea what Alice has done on her end, or indeed if she has done anything at all. There are now three cases:
1. Alice has performed her measurements, and by pure chance Bob has selected the same set of particles that Alice would have told him to select. I presume that in this case you would agree that this set of particles is still in state P. (The odds of this happening are, of course, vanishingly small, but still >0, and this is a thought experiment.)
2. Alice has NOT performed any measurements. Are Bob's particles still in state P?
3. Alice has performed measurements, and Bob's random selection does not match Alice's list of particles that don't require rotation. (This is the overwhelmingly likely scenario, of course.) In this case, Bob's particles are not in state P (because 3/4 of them require a rotation). But are they in some other pure state Q?
The answer to 2 and 3 must be "yes" if you want to avoid FTL (because otherwise Bob could determine whether or not Alice has performed measurements). So the conclusion is: if you randomly select N/4 particles from a mixed population, the resulting particles are all in some pure state. Do you agree with this? If so, were those particles in that same pure state before they were selected?
The conclusion I'm driving towards is that purity of state has nothing to do with the actual physical situation, it's a matter of perspective. Any physical state can appear to be pure if you look at it in the right way, but finding the "right way" to look requires additional information, i.e. information from Alice. Nothing Alice does changes anything about the physical situation on Bob's side. It simply produces the information Bob needs in order to look at his particles in the "right way" to see a particular pure state.
I thought you were referring to the relativity of simultaneity. I'm glad you insist on the FTL communication issue, maybe now I will be able to convince you that there is no issue at all...
> The answer to 2 and 3 must be "yes" if you want to avoid FTL (because otherwise Bob could determine whether or not Alice has performed measurements).
The answer to 2 is "no". If Alice has not performed any measurement, the N "B" particles that Bob has are not in a pure state. They are a subsystem of the N (indentical) entangled pairs formed by the N "A" particles that Alice has plus the corresponding N "B" particles that Bob has (let's call that state E).
So, depending on whether Alice has done the measurement or not, what we have is (I undertstand you don't really accept this description, it's for the sake of the argument where you will try to show how this description is unacceptable):
Measurement not done
Pairs remain entangled
The N particles held by Bob are not in a pure state
(The A/B pairs are in a pure state E)
Measurement done
Pairs are no longer entangled
The N particles held by Bob are each in one definite state P or P' or P'' or P'''
(Alice knows the precise state of Bob's particles, she knows how to rotate each particle to get the state P)
You say that, before any message from Alice can reach Bob, he can determine wheter or not she has performed the measurement.
> I thought you were referring to the relativity of simultaneity.
Yes, I was.
> I'm glad you insist on the FTL communication issue,
It amounts to the same thing.
> maybe now I will be able to convince you that there is no issue at all...
Maybe, except that I'm not sure you're clear on what we're actually disagreeing about.
> You say that, before any message from Alice can reach Bob, he can determine wheter or not she has performed the measurement.
Only in a hypothetical case, not realizable in practice (probably -- see below), where Bob has chosen N/4 particles at random and those just happen to coincide with the particles that Alice measured and found to require no rotation. In that case, Bob will, on your view, observe first-order interference if Alice has measured, and not otherwise.
Maybe it would help if I describe it as a detailed thought experiment. The experiment takes place in four phases.
Phase 1: Alice and Bob choose a basis for a pure state P which they will use for the remaining two phases.
Phase 2: Alice teleports N copies of state P to Bob using the normal protocol, including the transmission of the results of her measurements. Bob takes the N/4 particles that require no rotation and tests them to see if they produce first-order interference relative to the basis of P. I trust you and I agree that he will observe interference in this case.
Phase 3:
This is the part that can't be realized in practice, and I have to number the steps here because this phase runs in a loop. Phase 3 is designed to answer the question: what would have happened in phase 2 if Bob had chosen the same N/4 particles by pure chance? So...
1. N EPR pairs are distributed.
2. Alice performs her part of the teleportation protocol, but does NOT tell Bob the results of her measurements.
3. Bob selects N/4 particles at random and tests for first-order interference relative to P. (The vast majority of the time the result will be negative.)
4. Bob and Alice compare Bob's random selections in step 3 with the results of Alice's measurements in step 2. If they happen to coincide, i.e. Bob just happened to pick the correct N/4 particles that did not require rotation, the experiment halts, with the result being whatever Bob observed on the last iteration of step 3.
I trust that you and I will agree that Bob will also observe interference in this case.
Phase 4 (this is the interesting one):
This is identical to phase 3, except that steps 2 and 3 are reversed, i.e. Bob makes his random selection and interference measurement BEFORE (in a fully relativistic sense) Alice makes her measurements. i.e.:
1. N EPR pairs are distributed.
2. Bob selects N/4 particles at random and tests for first-order interference relative to P.
3. Bob communicates to Alice that he is done.
4. AFTER Alice receives Bob's signal from step 3, she performs her side of the teleportation protocol for N copies of P.
5. They compare the results of Alice's measurements in step 4 with Bob's random choices made in step 2. If they coincide (i.e. Bob just happened to choose the correct N/4 particles that did not require rotation) the experiment concludes, with the result being whatever Bob observed on the final iteration of step 2. Otherwise, they go to step 1 and try again.
On your view, the predicted result of phase 4 will be negative. So if Alice cheats and performs her measurements before receiving Bob's signal, Bob can tell.
Postscript: now that I think about it, it might actually be possible to perform this experiment. A mach-zehnder interferometer can be pretty sensitive, so it might be possible to get N down low enough that the experiment could terminate in a reasonable amount of time.
> I'm not sure you're clear on what we're actually disagreeing about.
To be clear, I disagree with your claim that the QM description is not correct because it leads to FTL communication.
If you stand by that, you're wrong. If that's not what you meant, then there is no disagreement.
Now, if I understand your latest comment, you say that it's "Only in a hypothetical case [...] where Bob has chosen N/4 particles at random and those just happen to coincide with the particles that Alice measured."
This is like saying that two standard 52-card decks can be used for FTL communication, in the hypothetical case where Bob picks the same card that Aliced picked. If you think that is FTL communication I cannot agree with that. This kind of "FTL communication" doesn't present any problem at all for the QM description.
Maybe I'm missing something, but I don't see in your description of the experiment any trace of potential FTL communication. You're not even trying! I would like to see something like the following (but I understand you cannot provide it, because it's impossible):
1) N EPR pairs are distributed
2) Bob does something ("selects N/4 particles at random and tests for first-order interference relative to P" or whatever) and concludes that "Alice has done something to her particles" or "Alice has not done anything to her particles" depending on the result of his measurement.
[The prediction is done at this point, the verification step follows.]
3) Alice tells Bob if she did or not measure her particles before he did. Or Bob tells Alice his prediction. Or Alice and Bob tell Charlie and he performs the check. In any case, this requires slower-than-light communication.
This experiment can be repeated several times to see if Bob's prediction is right every time or if at least there is some correlation between what Alice did and what Bob guessed.
Your description has additional steps involving communication between Alice and Bob before there is any result or prediction. It is not in any way incompatible with the standard QM formulation that you reject, as far as I can see.
It looks like if, in the card-picking example, Bob picks a card and then compares his card with Alice's card. If the card is the same they announce that FTL communication had been achieved. If the card is not the same, they try again. If you think this is not a fair characterisation, please make clear what prediction is Bob doing before communicating with Alice.
To make my point as clear as I can once again: according to QM, Bob cannot tell if Alice has performed or not a measurement on her side. For any measurement that Bob can do on his particle, the predicted distribution of outcomes conditional on "the entanglement is broken, the particle is now in an unknown pure state" is exactly the same as the predicted distribution of outcomes conditional on "the entanglement is intact, the particle is not in a pure state (the pair A/B is still in the pure state E)". Whatever the outcome he gets out of his measurement, it cannot be evidence for one or the other.
> your claim that the QM description is not correct
That's not my claim. My claim is that the change in the mathematical description of Bob's particles (from mixed states to pure states) when Alice measures her particles does not correspond to any actual physical change in Bob's particles.
We agree about the math, and we agree about the outcomes of all experiments performed by Alice and Bob.
There is one thing we disagree about, and that is the outcome of the thought experiment I described above (which may in fact be realizable in practice). I predict that you will observe "retrocausality" in phase 4, i.e. when the experiment halts, Bob's choices will correspond with Alice's measurements with (essentially) 100% certainty. (Of course, this isn't really retrocausality, but it appears that way.) AFAICT, on your view of the world, you would predict (essentially) a 0% chance of Bob's choices corresponding with Alice's measurements.
Just out of curiosity, are you a physicist? Because you obviously know what you're talking about.
BTW, just in case it wasn't obvious, I left out a detail in the description of the thought experiment: In phase 3, if the experiment does not halt at step 4 it loops back around to step 1, just as it does in phase 4.
>> your claim that the QM description is not correct
> That's not my claim. My claim is that the change in the mathematical description of Bob's particles (from mixed states to pure states) when Alice measures her particles does not correspond to any actual physical change in Bob's particles.
Well, according to quantum mechanics there is a physical change when Alice measures her particles because:
before Alice's measurement => the pair "A/B" is entangled
after Alice's measurement => the pair "A/B" is not entangled
According to you alternative description, is the pair A/B entangled or not after Alice's measurement? Or is entanglement not an actual physical property of the pair?
I am (or used to be, for a brief time many years ago) a physicist. We had the opportunity to discuss in person last week. I enjoyed that (and this) discussion, thanks. By the way, I said I would send you a comment from David Mermin about the recent paper from Frauchiger and Renner. Actually I was thinking about this article from Jeffrey Bub: https://arxiv.org/abs/1804.03267
> Well, according to quantum mechanics there is a physical change when Alice measures her particles because:
> before Alice's measurement => the pair "A/B" is entangled
> after Alice's measurement => the pair "A/B" is not entangled
No. That's the whole point. The presence of absence of entanglement is a function of the perspective you choose to take when modeling the situation, not a function of the actual physical situation.
Thanks for the pointer to the Bub paper, that's a great reference.
Maybe we should move this discussion to email? It's getting pretty deeply nested and I don't think anyone else is paying attention.
If you agree that they are all the same, then what you are saying is exactly the same as what I am saying, except for the "no entanglement" part. Whether or not you have entanglement or a mixed state depends on what point of view you choose to take. If you have an EPR state, then the joint system is in a pure state, but each individual particle is in a mixed state when viewed as an isolated system. That is what a mixed state is. A mixed state is nothing more or less than the state of a proper subset of an entangled system.