Measuring entangled particles along several angles

[Xilor]

Heya, I was wondering what happens when you measure the spin of particles along several different angles on an entangled pair.

As far as I understand Bell's theorem, it basically says that if you first measure particle A in some angle, and then measure a particle B at some some angle, you'll get a result at B that depends on which angle you chose at A. So if you picked a different angle A', we would have a different amount of expected correlations.

So measuring in this order:
AB
Gives different results at B than
A'B

Now I'm curious what happens if you measure A twice along different angles, in orders:
AA'B or A'AB
Out of these two, which of these would have the same results (on average) at B as measuring in order AB ? Is it one of them, both, neither? Or is this just a dumb question to begin with?

And as a follow up, if you do:
ABA'B or A'BAB does this end up giving you the same final result at B as the previous two orders?
How about ABB'B and AB'B? Are these the same as AB?

 

[stevendaryl]

After measuring the spin of either particle of an entangled pair, they are no longer entangled, so subsequent measurements of one particle are uncorrelated with measurements of the other.

 

[Xilor]

Ah alright, thanks. That's kind of interesting, that it 'breaks the connection' upon measurement but still allows the measurement to affect the other side before it breaks off.

What would happen if you perform the spin measurements on A and A' simultaneously/extremely close together in time? Is it known what happens in that case?
I don't really know how the spin measurement process works, but I can imagine it may be possible to arrive in a situation where it's kind of ambiguous whether the measurement on A or the one on A' happened first.

 

[stevendaryl]

Ah alright, thanks. That's kind of interesting, that it 'breaks the connection' upon measurement but still allows the measurement to affect the other side before it breaks off.

What would happen if you perform the spin measurements on A and A' simultaneously/extremely close together in time? Is it known what happens in that case?
I don't really know how the spin measurement process works, but I can imagine it may be possible to arrive in a situation where it's kind of ambiguous whether the measurement on A or the one on A' happened first.

I don't know whether that has been tested with two measurements very close together, but the prediction of QM is that it doesn't matter how close together they are measured. If you have two particles that are entangled, the first measurement of one particle will be correlated with the first measurement of the other, but subsequent measurements of either particle are uncorrelated.

 

[Xilor]

I don't know whether that has been tested with two measurements very close together, but the prediction of QM is that it doesn't matter how close together they are measured. If you have two particles that are entangled, the first measurement of one particle will be correlated with the first measurement of the other, but subsequent measurements of either particle are uncorrelated.

Does QM say anything about the duration of a measurement? Is it just instant? Is it expected to be one timestep if there's discrete time? Or could it be several timesteps or a range in continuous time?
I'd be interested to see what happens if you start a second measurement while an earlier one is still ongoing, if such a thing is possible at all. Or what happens when it's ambiguous what is 'first' because of the relativity of simultaneity thing.Oh, and how about a situation with more than 2 entangled particles? Suppose you have particles A,B and C. Could measuring: ABC give different results than ACB? And how about ABC vs. AB'C ? Or does it all just get unentangled the moment you measure A?

 

[Nugatory]

What would happen if you perform the spin measurements on A and A' simultaneously/extremely close together in time? Is it known what happens in that case?

. That's actually one of the most common cases. If your two detectors are equidistant from your source of entangled pairs, both particles will reach their detectors at about the same time.

I don't really know how the spin measurement process works...

Google for "Stern-Gerlach device". However, for practical reasons, entanglement experiments are more often done with polarized photons instead of particle spins.

I can imagine it may be possible to arrive in a situation where it's kind of ambiguous whether the measurement on A or the one on A' happened first.

Again, that's the most common experimental setup. Are you familiar with the concept of relativity of simultaneity?

When we said above that "both particles reach their detectors at about the same time" we were using the frame in which the lab is at rest. Using any other frame, these events will not be simultaneous - which happened first is inherently ambiguous (although "undefined" or "frame-dependent" might be better words).

 

[Xilor]

. That's actually one of the most common cases. If your two detectors are equidistant from your source of entangled pairs, both particles will reach their detectors at about the same time.

Hmm, I think there's a bit of confusion here. If you're talking about two detectors, then in the notation I used, you'd be talking about measurements on A and B. Not on A and A'. I meant to ask what simultaneous measurements on two angles of the same particle would end up doing. So that based on what you're seeing at B, you could tell which of A / A' actually ended up happening first.

Again, that's the most common experimental setup. Are you familiar with the concept of relativity of simultaneity?

Yes, I mentioned it in the post above. I'm actually a little confused what all of this means for relativity of simultaneity, and am trying to learn how the two work together here.Like, following my earlier thoughts, I'm coming to a conclusion which seems to break things, which probably some betrays ignorance on my part. I'd love to hear where I'm going wrong.

I see two options in the case of a 3 particle entanglement, I imagine QM already predicts one of them or something else entirely.
We can measure particles A,B and C. Let's say that measurement: ABC means that we first measure A, then B, then C, and these are all measured using the same predefined angle . Then if we do a measurement A',B' or C', we measure those same particles but in an angle perpendicular to the other angle.

The two options I see are:
Option 1: The observed results at C are only influenced by the first measurement you do. So if you do: AB'C you'll get different results at C than if you do B'AC. In the case of AB'C, since A and C are along the same angle, C should correlate with A. While in case of B'AC, they shouldn't correlate.

Option 2: The observed results at C are only influenced by the last measurement you do. If you do AB'C and B'AC you'll still get different results. Additionally, there's also going to be different results observed at C between doing AC and AB'C Now regardless of whether option 1 or 2 is real, it seems to get funky with relativity of simultaneity. Suppose you move the particles far away from each other using spaceships, and instead of just using one entangled set of particles, you use a bunch of them to be able to do statistical analysis, you also bring high precision clocks along, which were synchronized when everyone was together.
Now at a predetermined time on the clocks. We're going to measure A and B' on all particle-sets. At a time known to be later, we measure C.
Since there is a measurable difference between doing AB'C and B'AC, and since we're going to get some measurement at C, we should be able to tell in retrospect whether we had measured the particles labeled A before the particles labeled B.
This seems to not really play nice with relativity of simultaneity. Based on this, we'd have a way of telling us whether the spaceships carrying the A particles experienced more time dilation than the spaceship carrying the B particles, which shouldn't be possible.And if option 2 would be true, you seem to be able to do instant communication too because you could determine whether you're going to measure B' based on what you measure locally on A, before you have someone on the other end of the line measure C.

 

[Nugatory]

Hmm, I think there's a bit of confusion here. If you're talking about two detectors, then in the notation I used, you'd be talking about measurements on A and B. Not on A and A'. I meant to ask what simultaneous measurements on two angles of the same particle would end up doing. So that based on what you're seeing at B, you could tell which of A / A' actually ended up happening first.

I'm sorry, I did misunderstand what you're asking.
It's not possible, even in principle, to perform simultaneous measurements of the direction of a particle's spin along different axes. A measurement of A requires interacting with the particle in a way that leaves it in an eigenstate of A; a measurement of A' requires interacting with the particle in a way that leaves it in an eigenstate of A'; a simultaneous measurement would require interacting with the particle in a way that leaves the particle in a state that is an eigenstate of both A and A' - and there is no such state. (Observables like these that cannot be measured simultaneously because they have no common eigenstate are called "incompatible" or "non-commuting").

 

[Xilor]

I'm sorry, I did misunderstand what you're asking.
It's not possible, even in principle, to perform simultaneous measurements of the direction of a particle's spin along different axes. A measurement of A requires interacting with the particle in a way that leaves it in an eigenstate of A; a measurement of A' requires interacting with the particle in a way that leaves it in an eigenstate of A'; a simultaneous measurement would require interacting with the particle in a way that leaves the particle in a state that is an eigenstate of both A and A' - and there is no such state. (Observables like these that cannot be measured simultaneously because they have no common eigenstate are called "incompatible" or "non-commuting").

Alright, so theoretically impossible. But what would occur if you attempted this in practice? If it can't be put in a hybrid state, it should be put into one of them (or maybe flip back and forth?). What would determine which of the possible states ends up becoming relevant for the entangled partner? If the answer is just time, how would this work with relativity of simultaneity?
Is it just impossible to make a device that does a double measurement on one particle at spacetime coordinates that aren't connected by a light cone?

 

[Nugatory]

I'm actually a little confused what all of this means for relativity of simultaneity, and am trying to learn how the two work together here
...
I see two options in the case of a 3 particle entanglement
...
Since there is a measurable difference between doing AB'C and B'AC, and since we're going to get some measurement at C, we should be able to tell in retrospect

The answer here is pretty much the same as for the two-particle case, so you may want to take a look at some of the many threads on why we cannot communicate using two-particle entanglement. The quick summary is that the measurable differences only become apparent when all the separated observers get together and compare notes; anyone observer's results in isolation are statistically indistinguishable from the results they would get if they were the only one making measurements. When they do get together and compare notes they do find the statistical correlations that show that they were dealing with entangled particles. However, these correlations are the same no matter what order the measurements were made in, so there is no issue with relativity of simultaneity.

 

[Xilor]

The quick summary is that the measurable differences only become apparent when all the separated observers get together and compare notes; anyone observer's results in isolation are statistically indistinguishable from the results they would get if they were the only one making measurements.

Yes, I'm aware of that, I don't see how instant communication is possible if option 1 is right (though it seems possible in option 2). This is more about the apparent ability to be able to tell in retrospect which event objectively happened before another event based on the amount of correlations. This knowledge seems to break things.
I can't really tell if you're trying to say that this knowledge is unobtainable, or that this knowledge actually doesn't harm anything.

..However, these correlations are the same no matter what order the measurements were made in..

With this, do you mean that if you did measurements:
AB'C you'd get the exact same expected amount of correlations between A and C as you would if you did B'AC? How would that work?
Since A and C are parallel, and B' and C are perpendicular. Shouldn't one of them give full correlation, while the other has a 50% correlation?

 

[Nugatory]

Alright, so theoretically impossible. But what would occur if you attempted this in practice?

You'd have to specify the details of your measurement device before anyone can say what it would do, but we can confidently say that it will not deliver simultaneous measurements of both A and A'. For a concrete example, consider how we measure spin along a particular axis with a Stern-Gerlach device: we pass the particle through an inhomogeneous magnetic field oriented along the axis we're measuring, and see if the particle is deflected in the direction the field points (spin up) or opposite to it (spin down). But there's no such thing as a magnetic field that points in two directions at once, so no way of measuring the spin along two axes simultaneously.

The situation here is somewhat similar to what we have with perpetual motion machines: finding the problem in any particular complicated design may be a lot of work, but we already know from conservation of energy that the problem is there.

Is it just impossible to make a device that does a double measurement on one particle at spacetime coordinates that aren't connected by a light cone?

If you're talking about a single particle, everything you do to it must be inside a common light cone because it's following a time-like worldline so I'm not sure what you're asking here.

 

[Xilor]

You'd have to specify the details of your measurement device before anyone can say what it would do, but we can confidently say that it will not deliver simultaneous measurements of both A and A'. For a concrete example, consider how we measure spin along a particular axis with a Stern-Gerlach device: we pass the particle through an inhomogeneous magnetic field oriented along the axis we're measuring, and see if the particle is deflected in the direction the field points (spin up) or opposite to it (spin down). But there's no such thing as a magnetic field that points in two directions at once, so no way of measuring the spin along two axes simultaneously.

The situation here is somewhat similar to what we have with perpetual motion machines: finding the problem in any particular complicated design may be a lot of work, but we already know from conservation of energy that the problem is there.

Thanks, that makes sense. On the currently existing measuring devices it definitely doesn't seem possible, and I obviously can't fathom anything that could do simultaneous detection either. So I guess I'll just accept that this almost certainly just isn't possible.

If you're talking about a single particle, everything you do to it must be inside a common light cone because it's following a time-like worldline so I'm not sure what you're asking here.

That's what I suspected, but wanted a confirmation for, since my suspicions aren't always right. It could've been the case that you could make a measurement that gives you a particles spin at a spacetime position that doesn't equal the particles position. Like perhaps the particle could've had side-effects on nearby other particles due to its spin, so that you could do an indirect measurementDo you happen to have an answer for the questions in my other post as well? I'm unable to figure out the reason whether reality is like option 1,2, or something else based on other discussions about Bell's theorem. And why if reality is like options 1 or 2, that couldn't be used to obtain normally inaccessibly information about simultaneity. (or instant communication in case of the unlikely option 2, which would seem to allow the ability to influence what the other gets to see)

 

[Mentz114]

The two options I see are:
Option 1: The observed results at C are only influenced by the first measurement you do. So if you do: AB'C you'll get different results at C than if you do B'AC. In the case of AB'C, since A and C are along the same angle, C should correlate with A. While in case of B'AC, they shouldn't correlate.

Option 2: The observed results at C are only influenced by the last measurement you do. If you do AB'C and B'AC you'll still get different results. Additionally, there's also going to be different results observed at C between doing AC and AB'C

Two entangled things share the same same wave function so anything that changes this affects both particles. There is no concept of individual particles in the state so questions about precedence have no answer. It is not observable nor significant.

 

[Xilor]

Two entangled things share the same same wave function so anything that changes this affects both particles. There is no concept of individual particles in the state so questions about precedence have no answer. It is not observable nor significant.

But the angle of measurements used determines the amount of correlation you end up seeing if I understand Bell's Theorem correctly.
If you do AB'C, you start with a different angle of measurement than if you do B'AC. In the former, you start with an angle parallel to C, in the latter you start with something perpendicular to C.
Because different measurement angles are used, shouldn't you see different correlation patterns?

You should see different correlation patterns between AC and A'C. Why not between AB'C and A'BC ? (which if they're all identical should have the same results as AB'C and B'AC)

 

[Mentz114]

But the angle of measurements used determines the amount of correlation you end up seeing if I understand Bell's Theorem correctly.
If you do AB'C, you start with a different angle of measurement than if you do B'AC. In the former, you start with an angle parallel to C, in the latter you start with something perpendicular to C.
Because different measurement angles are used, shouldn't you see different correlation patterns?

You should see different correlation patterns between AC and A'C. Why not between AB'C and A'BC ? (which if they're all identical should have the same results as AB'C and B'AC)

I don't know what correlations you mean. When one of a pair of entangled particles ( say photons) is projected into a new angle by a polarizer the partner is also projected because the wave function changes. Also the entanglement is ended. You've been told 3 times now, so I think you should apply the rules and answer your own question.

 

[Nugatory]

With this, do you mean that if you did measurements:
AB'C you'd get the exact same expected amount of correlations between A and C as you would if you did B'AC? How would that work?
Since A and C are parallel, and B' and C are perpendicular. Shouldn't one of them give full correlation, while the other has a 50% correlation?

We have to specify the entangled state that we're preparing our groups of three particles in, and then work through the math.

Some notation: let's say that ##|uuu\rangle## is the state in which a measurement of any of the three particle spins will yield with certainty the result spin-up at a given angle; ##|uud\rangle## is the state in which these measurements will yield spin-up for particles A and B but spin-down for particle C; and so forth. We'll use a prime sign when we're talking about the result at a different angle so ##|ud'u\rangle## is the state in which a measurement of A or C at our original angle will yield spin-up with certainty while a measurement of B on the other angle will yield spin-down (and therefore a measurement of B on the original angle can yield either spin-up or spin-down with probabilities determined by the difference between the two angles).

An example of an entangled state might be ##\frac{1}{\sqrt{2}}(|udu\rangle+|dud\rangle)##. The value of all three spins are random, any measurement of any particle will yield up or down with equal probability, but they are correlated; when our observers compare notes they will find that A and C always have the same spin, and it is opposite from B.

If we measure A first, half the time we will get spin-up and the wave function will collapse to ##|udu\rangle## and the other half of the time we will get spin-down and the wave function will collapse to ##|dud\rangle## as we record a spin-down result. Clearly if we then measure B or C we have a 50% probability of getting either spin-up or spin-down, but either way the correlation will be respected - the times that C was spin-up are the times that A was also spin-up and vice versa, and likewise for B. Equally clearly it will turn out the same way if we measure B first.

But suppose we choose to measure B on an axis perpendicular to our original axis? We have ##|udu\rangle=|uu'u\rangle+|ud'u\rangle## and ##|dud\rangle=|dd'd\rangle-|du'd\rangle## so we rewrite our wave function using these identities to get ##\frac{1}{4}(|uu'u\rangle+|ud'u\rangle+|dd'd\rangle-|du'd\rangle)##. You can see for yourself what this collapses to after a measurement of A and then B' and compare with a measurement of B' and then A. You will find that the correlations are preserved, any individual measurement is completely random, and the order of the measurements makes no difference.

(If you choose an angle other than perpendicular for the B' measurement, the algebra gets a bit messier but it still comes out the same way).

 

[Xilor]

I don't know what correlations you mean. When one of a pair of entangled particles ( say photons) is projected into a new angle by a polarizer the partner is also projected because the wave function changes. Also the entanglement is ended. You've been told 3 times now, so I think you should apply the rules and answer your own question.

When I try to apply the rules I'm told, I can't do anything but get to a situation where you can break relativity of simultaneity. I've tried to express this several times but am not being engaged on my point of confusion. I'm not receiving the key piece of information I need to stop making whatever error I'm making.

Let me try to write it out again, maybe it's easier to see where I'm going wrong then. Please actually follow along and indicate where I'm making my errorSuppose you make a set of triply entangled particles (not 3 pairs but a triplet). If you call the particles in this triplet A,B and C, then I expect these things to happen:

1. If you measure A along the Z-Axis, and then measure C along exactly the same angle, because they are 'the same particle', you should see that the measurement of C should be equal to that of A.

2. If you measure A along the Z-Axis, and then measure C along the X-Axis, because they are the same particle, and because spin needs to be quantized, and because of the way QM works, you should see that there's a 50% chance that your result at C equals the one you got at A.

3. Since A is identical to B, the last two statements can be repeated for the combination B->C (as well as A->B, B->A, C->A, C->B) without changing the results.

4. If you measure A along the Z-Axis, then measure B along the X-Axis, and then measure C along the same Z-Axis angle as you measured A. Then according to the rules as I understand them, entanglement was already broken when A was measured. So your measurement of B has no effect on C (or A). Therefore, just like in 1. A and C are perfectly correlated. B however only has a 50% chance of being correlated with both A and C.

5. If you measure A along the X-Axis, then measure B along the Z-Axis, and then measure C along the same Z-axis angle as you measured B. Then since entanglement was broken after A was measured, the measurement on B has no effect on C. Since we measured A along an angle perpendicular to C, A and C should only be correlated 50% of the time. B and C are not necessarily correlated either, since they are no longer entangled and there are no hidden variables, both should have an independent 50% chance of being correlated with A.

6. Since all particles are equal, and labels are just labels, we'd get the same result as 4. and 5. if we just flip the A and B labels whenever mentioned.

7. In 4. we measured A(Z) -> B(X) -> C(Z) with full correlation between results at A and C. Through 5. and 6. we know that if we did: B(X) -> A(Z) -> C(Z), we wouldn't necessarily have a full correlation between A and C.

8. The two ways of doing measurements mentioned in 7. are identical except for the way the measurements of A and B are ordered. If we agree to not communicate in advance whether we'll do the measurement on A or B first, then we can after the experiment still find out whether A or B happened first. If A happened first, A and C should correlate 100%. If B happened first, A and C should correlate 50% of the time. All we'd need to do is execute the same experiment multiple times simultaneously

9. Since we can figure out whether A or B happened first, we run into issues with simultaneity. The way this information about time-ordering was gathered was frame-invariant, that shouldn't be possible.

 

[Xilor]

But suppose we choose to measure B on an axis perpendicular to our original axis? We have ##|udu\rangle=|uu'u\rangle+|ud'u\rangle## and ##|dud\rangle=|dd'd\rangle-|du'd\rangle## so we rewrite our wave function using these identities to get ##\frac{1}{4}(|uu'u\rangle+|ud'u\rangle+|dd'd\rangle-|du'd\rangle)##. You can see for yourself what this collapses to after a measurement of A and then B' and compare with a measurement of B' and then A. You will find that the correlations are preserved, any individual measurement is completely random, and the order of the measurements makes no difference.

Aaah, thank you so much for the write up! I think I get it now, this is why I was wrong (please correct if still wrong):

My statement at the end of 5. in my previous post was incorrect, leading to the false conclusion.
As part of the collapse made possible through a measurement of A along the X-axis, it actually also sets the results we will get in the future if we measure along the Z-axis for B and C.
So basically, it gives B and C some kind of 'memory' that allows them to know what values to take upon being measured along any other angle, we just don't know yet what these will be, so it'll still seem random.

Does this information persist? Or does it just last till the first time you measure them? Like suppose you get A(X) = down, and B(Z) = up, C(Z) = up, if you then later on go and measure B among the X and Y axes, will a later measurement along the Z-axis still return up?

 

[DrChinese]

Suppose you make a set of triply entangled particles (not 3 pairs but a triplet). If you call the particles in this triplet A,B and C, then I expect these things to happen...

It doesn't work as you imagine. There are conservation rules at work in most entanglement scenarios (2 or more). Yes, you can entangle 3 particles and there will be conservation of total quantities (such as spin, momentum, etc.). But don't think of the 3 component particles as clones of each other, as they are not by a lot shot. Because there are 3, the math is quite different. If you learn the spin of A, you can deduce the spin of (B+C) combined but you learn nothing about B or C individually.

Honestly, you should skip this line of thinking until you have a better grasp of 2 particle entanglement.

 

[Xilor]

Honestly, you should skip this line of thinking until you have a better grasp of 2 particle entanglement.

Alright, that's fair.
Thanks for your help everyone! I learned a ton

 

[Nugatory]

My statement at the end of 5. in my previous post was incorrect, leading to the false conclusion.

The problem is one step earlier:

4. If you measure A along the Z-Axis, then measure B along the X-Axis, and then measure C along the same Z-Axis angle as you measured A. Then according to the rules as I understand them, entanglement was already broken when A was measured.

Measuring A breaks the entanglement between A and the rest of the system, but it doesn't disturb any other entanglements that might be present in the rest of the system. Consider, for example, the entangled state ##|uud\rangle+|udu\rangle+|duu\rangle+|ddd\rangle## (I cannot imagine any physical process that would lead to this state, but it's valid state that we can treat according to the Rules of Quantum Mechanics if we could find a way of preparing it). Now a measurement of A leaves B and C entangled - but try it yourself, don't take my word for it! Look carefully at the two possible post-measurement states to see what makes them entangled, compare with the two possible states of my example in post #17 after a measurement of A, and you'll what "breaking entanglement" really means.

So basically, it gives B and C some kind of 'memory' that allows them to know what values to take upon being measured along any other angle, we just don't know yet what these will be, so it'll still seem random.

That is a really bad way of thinking about it, and suggests that you haven't picked up on what @Mentz114 is saying: "There is no concept of individual particles in the state". You should not be thinking of this quantum system as three particles; you should be thinking of the quantum state as a single object that has three measurable properties (result at detector A, result at detector B, result at detector C).

 

[Nugatory]

you should skip this line of thinking until you have a better grasp of 2 particle entanglement.

Listen to this man.

 

[Xilor]

Measuring A breaks the entanglement between A and the rest of the system, but it doesn't disturb any other entanglements that might be present in the rest of the system. Consider, for example, the entangled state ##|uud\rangle+|udu\rangle+|duu\rangle+|ddd\rangle## (I cannot imagine any physical process that would lead to this state, but it's valid state that we can treat according to the Rules of Quantum Mechanics if we could find a way of preparing it). Now a measurement of A leaves B and C entangled - but try it yourself, don't take my word for it! Look carefully at the two possible post-measurement states to see what makes them entangled, compare with the two possible states of my example in post #17 after a measurement of A, and you'll what "breaking entanglement" really means.

Okay, so it's more of a filter of some of the possible entangled states than a hard break of an entangled connection? By measuring A(X), it's not forcing B(Z) or C(Z) into anything, but it only filtered out the possibility for B(X) and C(X) to be anything other than correlated with A(X). B(Z) and C(Z) can still be anything, as long as they're still correlated.

Are the possible states you mentioned chosen for any specific reason? Or are you just constraining the possible options a bit for clarity? I noticed you used a minus sign for one possibility in #17 and don't really understand why that's there.

 

[DrChinese]

By measuring A(X), it's not forcing B(Z) or C(Z) into anything, but it only filtered out the possibility for B(X) and C(X) to be anything other than correlated with A(X). B(Z) and C(Z) can still be anything, as long as they're still correlated.

For simplicity, suppose the total energy of entangled particles A, B and C is 100 (because of the conservation rule). You measure A to be 30 (and so it is no longer entangled with B and C on this basis). You now know that B+C=70, right? Any combination (assuming there are no other constraints) in which B+C=70 are viable. If you then measure B to be 21, you know C=49.

The same analogously applies to spin, momentum, etc.

 

[Nugatory]

Are the possible states you mentioned chosen for any specific reason? Or are you just constraining the possible options a bit for clarity?

In general the initlial spin state of the three-particle system can be any linear combination of the eight common eigenstates of A, B, and C (##|uuu\rangle##, ##|uud\rangle##, ##|udu\rangle##, ##|udd\rangle##, ..., ##|ddd\rangle##). I chose those particular examples because they conveniently illustrated the point that I was trying to make at the moment: The first is an easy way of showing that you get the same correlations no matter the order of the measurements, and the second is an an example of how measuring A can leave B and C entangled.

I noticed you used a minus sign for one possibility in #17 and don't really understand why that's there.

That sent me back to my earlier post looking for the minus sign that you're talking about, and I noticed some smallish errors.
##|udu\rangle=|uu'u\rangle+|ud'u\rangle## is supposed to be ##|udu\rangle=\frac{1}{\sqrt{2}}(|uu'u\rangle+|ud'u\rangle)##
##|dud\rangle=|dd'd\rangle-|du'd\rangle## is supposed to be ##|dud\rangle=\frac{1}{\sqrt{2}}(|dd'd\rangle-|du'd\rangle)##

But with that said, I presume you're asking about the minus sign in the expression for ##|dud\rangle## because that's the only one I see, and yes, it is there for a reason. I can't come up with any good B-level explanation of why it has to be there; the best I can do is:
- Try Giancarlo Ghirardi's book "Sneaking a look at god's cards".
- If you think of ##|d\rangle## (we're only considering how we describe the parts of the wavefunction that affect the result of measurements of B) as a vector pointing north, ##|u\rangle## as a vector pointing east, ##|d'\rangle## as a vector pointing northeast, and ##|u'\rangle## as vector pointing northwest, those two identities are saying that a vector pointing north is the sum of a vector pointing northeast and a vector pointing northwest and a vector pointing east is what you get when you subtract a vector pointing northwest from a vector pointing northeast.

 

[Derek P]

Measuring A breaks the entanglement between A and the rest of the system, but it doesn't disturb any other entanglements that might be present in the rest of the system. Consider, for example, the entangled state ##|uud\rangle+|udu\rangle+|duu\rangle+|ddd\rangle## Now a measurement of A leaves B and C entangled - but try it yourself, don't take my word for it! Look carefully at the two possible post-measurement states to see what makes them entangled...

I'm confused. Looks to me like you get a mixture of |ud>+|du> and |uu>+|dd>. That's two different entanglements - one with perfect correlation and the other with perfect anticorrelation. I can't see any way that a measurement on B and C could reveal this situation. Certainly if one used the A measurement to sort the BC results, one would find the correlation between A and BC preserved, but to say B and C are themselves entangled doesn't make sense to me. It's very late and I may be missing something but could you explain please?

 

[Derek P]

Does QM say anything about the duration of a measurement? Is it just instant? Is it expected to be one timestep if there's discrete time? Or could it be several timesteps or a range in continuous time?
I'd be interested to see what happens if you start a second measurement while an earlier one is still ongoing, if such a thing is possible at all. Or what happens when it's ambiguous what is 'first' because of the relativity of simultaneity thing.

Not sure whether you got a complete answer to this but the order of measurement AB or BA is unimportant. The correlation will be the same. Don't confuse making the same measurement on two particles with the taking two different measurements on the same particle. In the latter case the order does matter.

Where order and simultaneity come into it is when you try to construct a causal model. QM per se does not tell you how events at A affect the probabilities at B, it just asserts that the correlation will be present. Saying how it happens goes beyond QM and is a matter of interpretation. Unfortunately interpretation is rather a dirty word in PF because it has always been associated with truly dreadful philosophy - Schroedinger called Bohr's theory of reality "a philosophical extravaganza born of despair in the face of a grave crisis" and things have not improved much since. But it is not unreasonable to ask that the things scientists talk about should actually exist. And interpretations like MWI do assert the reality of the wavefunction.

Do you see where this takes us? Experiments are constructed that make it impossible for the events at A to affect the measurement at B by means of a signal of any kind. Typically the measurements are simultaneous but so far apart that, even at light speed, nothing can get from one to the other before both measurements are done and dusted. And the experiments are such that, if quantum theory is correct, then the probabilities at B - the actual detection rate once B chooses an angle - will depend on what happens at A and vice versa. And they do. Common causes are ruled out by design so events at A must be causing something to happen at B. Yet there can be nothing passing between them to make it happen.

So where did interpretation creep into this simple physical but utterly paradoxical scenario? People often play fast and loose with causality trying to make sense of it all, but it is worth mentioning that at least one interpretation, MWI again, manages perfectly well without any such hocus-pocus. It avoids the paradox by eschewing the intuitive idea of definite events. Probabilities do not refer to random events, to nature choosing one possibility. Instead they refer to observed statistical frequencies in a massive superposition of all possible histories. The maths is non-trivial but resolution of the superposition - collapse of the wavefunction - is an illusion: each observer state sees a different collapse - Schroedinger's cat is alive in one phenomenal world (note the term, phenomenal, please!) and dead in another.

But that is MWI. You are working within a collapse paradigm, presumably some version of Copenhagen, which is regarded as orthodoxy. So, at the risk of sounding heartless, you will never "make sense" of entanglement. Events at A (if definite events exist) do affect detection rates at B even though it's physically impossible. We can chase the paradox around and say entanglement is a non-local physical phenomenon. Or we can say that temporal ordering of cause and effect is optional. Or that reality does not exist until all relevant observations have been made. It is all metaphysical philosophy and it is occasioned by the prior choice of interpretation - a collapse theory. In other words you have unwittingly assumed that a measurement at A actually collapses the wavefunction - which then raises the very questions you ask about when the "part of the wavefunction" at B is affected. Those here who take a formal mathematical approach will say that you can't separate the wavefunction that way. Which means you are forbidden to ask the question ("Shut Up And Calculate!").

But you can if you leave the wavefunction un-collapsed. No collapse means that there are no definite events (that is random events with real, definite outcomes) at A to affect B so the question of "when" or even "how" does not arise. But if you feel you must keep your collapse and your definite events then there is no physically plausible explanation. Indeed Bell's theorem is a direct proof that no such theory is even possible, whether based on quantum mechanics or not. So you are stuck with SUAC.

HTH