Solving Entanglement Q: EPR Paradox & Quantum Computing

[ballzac]

Hi all. I'm having trouble 'getting' entanglement. We were shown the EPR paradox in a lecture once, and I didn't get it. We've also been shown quantum computing, and there was something (can't remember the details) that required an understanding of entanglement, and I didn't get it. It's a little embarrassing to be at third year level and not get it.

I've been trying to figure it out by reading stuff about it. Here is one main aspect I don't understand:
http://en.wikipedia.org/wiki/EPR_paradox

Suppose Alice measures the z-spin and obtains +z, so that the quantum state collapses into state I. Now, instead of measuring the z-spin as well, Bob measures the x-spin. According to quantum mechanics, when the system is in state I, Bob's x-spin measurement will have a 50% probability of producing +x and a 50% probability of -x. Furthermore, it is fundamentally impossible to predict which outcome will appear until Bob actually performs the measurement.
Here is the crux of the matter. You might imagine that, when Bob measures the x-spin of his positron, he would get an answer with absolute certainty, since prior to this he hasn't disturbed his particle at all. But, as described above, Bob's positron has a 50% probability of producing +x and a 50% probability of -x—random behaviour, not certain. Bob's positron knows that Alice's electron has been measured, and its z-spin detected, and hence B's z-spin calculated, so its x-spin is 'out of bounds'.

I don't get why we would expect Bob to get an answer with absolute certainty when measuring x-spin. Prior to measuring Alice's z-spin, Bob's x-spin is unknown, and therefore + or - is equally likely. AFTER measuring Alice's z-spin, the same is true. So measuring Alice's z-spin has no effect on Bob's x-spin. Only on his z-spin.

If the z component of both Alice and Bob's particle are determined at creation - and opposite - then when we measure Alice's z-spin, we know Bob's. If we then measure Bob's x-spin, it will have a fifty fifty chance of being + or -. The way I have stated it, does NOT imply non-locality, therefore I am misunderstanding some aspect.

Can anyone explain what I am missing?

(EDIT: I should point out that I realize that it is understood that the spins are not determined on creation, but actually at measurement, I just don't understand how this conclusion has been arrived at. ie How does the observation that the x-component of B is random suggest that it 'knows' about the measurement made on A?)

(EDIT 2: Also, I don't understand what it means that B's x-spin is "out of bounds". It says that you can measure it and it has equal probability of being measured in each state. So how is it out of bounds? If you measure A's z-spin and then B's x-spin you know the x and z spin of both particles, violating HUP, so again I am missing something.)

(yet another EDIT:

From http://en.wikipedia.org/wiki/Quantum_entanglement

Now suppose Alice is an observer for system A, and Bob is an observer for system B. If Alice makes a measurement in the eigenbasis of A, there are two possible outcomes, occurring with equal probability:[citation needed]
Alice measures 0, and the state of the system collapses to .
Alice measures 1, and the state of the system collapses to .
If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1. If the latter occurs, (Alice measures 1) then Bob's measurement will return 0 with certainty. Thus, system B has been altered by Alice performing a local measurement on system A. This remains true even if the systems A and B are spatially separated. This is the foundation of the EPR paradox.

Suppose I put one red and one blue marble in a black felt bag. I shake the bag around to mix them up. Blindfolded, I remove one marble and put it in another black felt back. I give one bag to Alice. If I open my bag, I have a fifty-fifty chance of getting either red or blue. If, however, I ask Alice to look in her bag and she tells me she has a blue marble, I know with 100% certainty that my marble is red. I open my bag and, indeed, it is red.

Were the marbles entangled? Has the measurement on system A altered the outcome on system B? No. The outcome is the same as in the quote, but the interpretation is different. What am I missing?

 

[zonde]

It seems that wikipedia article is very poor and completely misses the point.

Let me start with this. If we are not very strictly sticking to Bell's theorem where spin entangled particles where used for derivation it would be logical to switch to photon polarization entangled states for explanation.
That's because there are no real experiments with spin entanglement but there are plenty of photon polarization entanglement experiments. That way we can check our reasoning with empirical results of experiments when we need.

Second, intuitive picture of some shared polarization angle for entangled pair fails not at angles where we get 100% chances (perfect correlation) versus 50% chances (no correlation) but at angles between these perfect correlation and no correlation angles.
For polarization it is: 0° and 90° - perfect correlation (+1 and -1); 45° - no correlation; and problematic angles usually used - 22.5° and 67.5°.
If you need then for spin that was: 0° and 180° - perfect correlation (+1 and -1); 90° - no correlation; and problematic angles usually used - 45° and 135°.

So in order to see the crux of the matter you have to compare what would be result for 22.5° and 67.5° relative angles (I refer to photon polarization) when you use your intuitive picture versus quantum predicted and experimentally verified result.

 

[nomadreid]

The closest source that I know of that gets as intuitive as possible (in a counter-intuitive topic) without sacrificing accuracy concerning entanglement and the EPR experiment is on pages 94-104 of "An Introduction to Hilbert Space and Quantum Logic" by David W. Cohen, published by Springer Verlag, 1989. It should be in your university library (if it isn't, go berate the librarian). That section requires very little mathematics, and the book requires no background in physics. The rest of this small book is a gem as well.

 

[DrChinese]

Hi all. I'm having trouble 'getting' entanglement. We were shown the EPR paradox in a lecture once, and I didn't get it. We've also been shown quantum computing, and there was something (can't remember the details) that required an understanding of entanglement, and I didn't get it. It's a little embarrassing to be at third year level and not get it.

I've been trying to figure it out by reading stuff about it. Here is one main aspect I don't understand:
http://en.wikipedia.org/wiki/EPR_paradox

I don't get why we would expect Bob to get an answer with absolute certainty when measuring x-spin. Prior to measuring Alice's z-spin, Bob's x-spin is unknown, and therefore + or - is equally likely. AFTER measuring Alice's z-spin, the same is true. So measuring Alice's z-spin has no effect on Bob's x-spin. Only on his z-spin.

If the z component of both Alice and Bob's particle are determined at creation - and opposite - then when we measure Alice's z-spin, we know Bob's. If we then measure Bob's x-spin, it will have a fifty fifty chance of being + or -. The way I have stated it, does NOT imply non-locality, therefore I am misunderstanding some aspect.

Can anyone explain what I am missing?

(EDIT: I should point out that I realize that it is understood that the spins are not determined on creation, but actually at measurement, I just don't understand how this conclusion has been arrived at. ie How does the observation that the x-component of B is random suggest that it 'knows' about the measurement made on A?)

(EDIT 2: Also, I don't understand what it means that B's x-spin is "out of bounds". It says that you can measure it and it has equal probability of being measured in each state. So how is it out of bounds? If you measure A's z-spin and then B's x-spin you know the x and z spin of both particles, violating HUP, so again I am missing something.)

(yet another EDIT:

From http://en.wikipedia.org/wiki/Quantum_entanglement
Suppose I put one red and one blue marble in a black felt bag. I shake the bag around to mix them up. Blindfolded, I remove one marble and put it in another black felt back. I give one bag to Alice. If I open my bag, I have a fifty-fifty chance of getting either red or blue. If, however, I ask Alice to look in her bag and she tells me she has a blue marble, I know with 100% certainty that my marble is red. I open my bag and, indeed, it is red.

Were the marbles entangled? Has the measurement on system A altered the outcome on system B? No. The outcome is the same as in the quote, but the interpretation is different. What am I missing?

Ok, you have made a very good start. You realized that that the z spin is not determined at start of entanglement but rather at measurement. Then you backtracked and asked about the marbles. That is good too, as you have distilled things to a good level to continue.

If the marbles analogy were good, then it would be a fact that you measure Alice (A) and Bob (B) and get information about both their x and z spins. Since those are non-commuting observables, this would violate the Heisenberg Uncertainty Principle (HUP). In essence, that was the EPR argument. So you are standing on pretty good ground so far. EPR claimed the HUP would fail in this case.

But nature is tricky. It would appear at first glance as if you could not demonstrate whether this would truly mean the HUP fails. But then Bell's Theorem came on the scene about 30 years after EPR. Bell noticed that x and z are independent components, but that mixtures of x and z would follow certain rules - specifically Malus. In other words, your marble example looks good for the "pure" case where the angles are 90 degrees apart for electrons (45 degrees for photons). But the analogy falls apart at other angles. If you are with me to this point I can explain this in more detail. You might want to look up Bell's Theorem on wiki too.

As zonde indicates, there are specific angles in which the issue becomes fairly obvious - usually photons are used for the example since that is what is used for most entanglement experiments.