3 Particle entanglement

[Isaac Hart]

TL;DR

Questions about distinguishing between unentangled and entangled states

Say you had 3 particles all of which were entangled in the GHZ format. What methods could you use to measure the entanglement between two of the particles? Say particle A was far away and particles B and C were close together. If you used magnetic fields at point A to break the entanglement between the 3 particles how could you detect this at points B and C, what methods could you use?

 

[Vanadium 50]

TL;DR Summary: Questions about distinguishing between unentangled and entangled states

You can stop right there. You can't look at a particle and say "yes, this particle is entangled" and "no, that one is not."

 

[Isaac Hart]

There must be some mechanism you can use to determine whether two particles are entangled or not—thats like the fundamentals of quantum mechanics. For instance, if one particle had a spin state of up and the other down, you would know they are entangled. And if you had two particles both up you would know no entanglement is present. I would like to know how you can determine entanglement in 3 particle entanglement systems.

 

[PeterDonis]

There must be some mechanism you can use to determine whether two particles are entangled or not

You didn't say two of the particles are entangled; you said that all three are entangled. If all three are entangled, it makes no sense to even ask whether two of them are entangled or not. Only all three together have a well-defined state.

 

[PeterDonis]

If one particle had a spin state of up and the other down, you would know they are entangled.

No, you don't. There are unentangled states of the two particle system where one particle is spin up and the other is spin down.

I think you need to think more carefully about what you are asking.

 

[PeterDonis]

if you had two particles both up you would know no entanglement is present.

This is wrong as well. You can have entangled states of two particles where both spins are the same.

Have you actually looked at the math of entangled states?

 

[pines-demon]

TL;DR Summary: Questions about distinguishing between unentangled and entangled states

Say you had 3 particles all of which were entangled in the GHZ format. What methods could you use to measure the entanglement between two of the particles? Say particle A was far away and particles B and C were close together. If you used magnetic fields at point A to break the entanglement between the 3 particles how could you detect this at points B and C, what methods could you use?

Let me get this straight you want to know if A was measured by looking only at B and C. Then the answers is that it is not possible, that would lead to faster-than-light signaling which is forbidden.

 

[Isaac Hart]

Please can someone just tell me what methods you can use to measure or detect the integrity of entanglement between 2 out of 3 entangled particles?

 

[pines-demon]

Please can someone just tell me what methods you can use to measure or detect the integrity of entanglement between 2 out of 3 entangled particles?

Can you write this mathematically? It would help a lot.

Edit: I mean, write the state and point what you want to know.

 

[Isaac Hart]

Can you write this mathematically? It would help a lot.

Edit: I mean, write the state and point what you want to know.

Okay so I would like to know if and how it would be possible to determine whether B and C are entangled or unentangled without talking to particle A. Thank you very much for offering to help, I am below highschool level so am not sure about the maths involved.

 

[PeterDonis]

Please can someone just tell me what methods you can use to measure or detect the integrity of entanglement between 2 out of 3 entangled particles?

Go read what I said in post #4, carefully. The question doesn't even make sense.

 

[PeterDonis]

If you used magnetic fields at point A to break the entanglement between the 3 particles

This description is interpretation dependent. In some interpretations, such as the MWI, measurement doesn't break entanglement, it just spreads it to include the measuring device. Discussions about what QM interpretations say belong in the interpretations subforum, not this one.

 

[PeterDonis]

@Isaac Hart your question as it stands is not answerable. It might help to have more background about why you are asking it. What are you trying to figure out? How did this question come up?

 

[DrChinese]

Okay so I would like to know if and how it would be possible to determine whether B and C are entangled or unentangled without talking to particle A. Thank you very much for offering to help, I am below highschool level so am not sure about the maths involved.

This helps me understand what you are asking. Of course, the first thing I would recommend (for the mathematical treatment) is simply looking at the Wikipedia page for GHZ state, which hopefully you’ve already done. The GHZ state is a fairly complicated thing to understand, a lot more complicated than the usual Bell state (2 particle entanglement).

As several have already mentioned, it is not possible to examine any two particles of a three particle state and then make the statement they are definitely GHZ entangled or not. You would normally expect to examine all three, where one of the three is observed on an orthogonal basis.

Further, normally, you cannot examine any entangled pair and make a statement on a single trial. It usually takes a statistical sample to arrive at any kind of reasonable conclusion.

 

[Vanadium 50]

Maybe it will help to think about this. I take N electrons from entangled pairs and put them in a box, keeping the other set here. I add M electrons to that box that are not part of pairs. I mail you the box and ask you to tell me N.

What do you do?

 

[Nugatory]

For instance, if one particle had a spin state of up and the other down, you would know they are entangled.

You do not. Even two random unrelated particles will show opposite results 50% of the time.

The only way of detecting entanglement is to measure both sides of many pairs, then get together after the fact and compare notes. If we find that every single time over a large number of pairs we had opposite results, then we conclude with high probability that our particle source is creating entangled pairs.
Once we have come to that conclusion we will use this source in our experiments, assuming that whenever we detect particles at both detectors at about the same time (after allowing for travel time between source and detector - the detectors are seldom exactly equidistant) we have observed at members of an entangled pair.

(Real experiments are complicated by random unrelated background particles that just happen to blunder into our detectors at about the same time, as well as many other sources of randomness. Even working with a known-good source of entangled pairs, detectors only a few tens of centimeters apart, I was never able to get 100% anti-correlation in my data).

 

[Nugatory]

Please can someone just tell me what methods you can use to measure or detect the integrity of entanglement between 2 out of 3 entangled particles?

We do what I described in the previous post, except with three detectors. We start with a particle source, we set up detectors at three locations A, B, and C, then we let the thing run for a while recording our results. Then we get together and compare notes from all three sites - if the right correlations are found between sets of three more or less simultaneous detections we conclude that our source is producing three-way entangled pairs.

 

[PeterDonis]

We do what I described in the previous post, except with three detectors.

That will, as you say, tell you about whether all three particles are being produced in an entangled state. But the OP asked about detecting "the integrity of entanglement between 2 out of 3 entangled particles". The method you describe will not detect that--nothing can, because the concept does not make sense.

 

[pines-demon]

I was thinking that OP was suggesting that we assume that we know that the three particles are prepared in the GHZ state $(|000\rangle+|111\rangle)/\sqrt2$. But even in that case:

• It is already trivial that the three particles are entangled, per assumption.
• You can figure out some weird basis to measure B and C as an entangled pair [e.g. measure in the 2 particle basis $(|00\rangle\pm|11\rangle)/\sqrt2$]. However that just forces B and C to be maximally entangled and separable from A.
• There is no way of telling if A has been measured or not by just measuring B and C (otherwise it would lead to faster-than-light communication).

 

[pines-demon]

The only thing that I think could be partially useful is that if you are not sure if the original state is a GHZ state. For example, if you measure B and C in the Bell basis, and you get a value corresponding to either $(|01\rangle\pm|10\rangle)/\sqrt{2}$ then your state was NOT a GHZ state, but if you get any of the other two states you cannot say anything conclusively.

 

[Isaac Hart]

This helps me understand what you are asking. Of course, the first thing I would recommend (for the mathematical treatment) is simply looking at the Wikipedia page for GHZ state, which hopefully you’ve already done. The GHZ state is a fairly complicated thing to understand, a lot more complicated than the usual Bell state (2 particle entanglement).

As several have already mentioned, it is not possible to examine any two particles of a three particle state and then make the statement they are definitely GHZ entangled or not. You would normally expect to examine all three, where one of the three is observed on an orthogonal basis.

Further, normally, you cannot examine any entangled pair and make a statement on a single trial. It usually takes a statistical sample to arrive at any kind of reasonable conclusion.

Apon doing research, I think I meant to say a W state entanglement. In this type of entanglement, complete loss of entanglement can be achieved by actions at A, which could be observable at points B and C. I would very much like to know, as I said, what methods you could use to determine quantum entanglement or not. I know measuring violations in Bell's theorem was one of them but I was wondering if you needed to do this between all 3 particles.

 

[Nugatory]

I would very much like to know, as I said, what methods you could use to determine quantum entanglement or not.

There is no way to determine that any particular particles are or ever have been entangled.

The best we can do is to assume that they are entangled because they were emitted by a source of entangled particles (or created by a process that entangles particles).
We determine that we have a such a source/process as described in posts #16 and #17 above.

 

[Nugatory]

I know measuring violations in Bell's theorem was one of them

If you have a setup that can detect violations of Bell's inequality then you already have an easier way of determining that your particle source is producing entangled particles: just measure all the particles in each pair (or triple if you're working with three-particle entanglement) on the same randomly chosen axis for many pairs/triples. Get enough measurements and you can conclude with high probability that the particles coming out of that source are entangled.

 

[pines-demon]

Repeating my statement for W state. With the assumption that you know that the particles A, B and C are in a W state $(|001\rangle+|010\rangle+|100\rangle)/\sqrt3$, then:

• It is trivial that the three particles are entangled, per assumption.
• You can measure B and C in the Bell basis, if you get the state $(|01\rangle+|10\rangle)$ then you forced your state to be maximally entangled, if you get something different the state is no longer entangled (half of the time I think).
• If A was measured or not, it is not possible to tell by just measuring B and/or C (faster-than-light communication is not possible)

If you do not assume that A,B,C are in a W state, and you have a single copy of A,B,C, then there are measurements that you can perform in B and C that can tell you that your state was NOT in a W state but there is no measurement that can confirm definitely that it was a W state.

If you want to confirm that it is a W state you need many copies of the state (a source of this state) and perform many measurements on the three particles until you have statistical confidence of your state (Bell tests do that).