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So I keep reading all these articles on the EPR paradox, and I follow them pretty easily right up until it gets to the most important matter.

Assuming you are trying to measure x and y spin,

Wikipedia and others say that when you measure x-spin on the first particle, it suddenly becomes impossible to measure the y-spin on the other particle.

But no one really goes on to say what this means in a physical sense.

Let's say you have 2 actual detectors. When the first particle hits the x-detector, now x-spin is measured for both particles. When the second particle hits the y-detector, now y-spin is measured for both particles. But all these articles say the second detector is unable to measure y-spin. So what happened? Did the detector just explode or something?!

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I looked through the Wikipedia article you linked to and I don't see any place where it says that it's impossible to measure the y-spin on the other particle. Can you cite that more specifically? – David Z Jul 28 '11 at 16:32
up vote 5 down vote accepted

I have to think David will agree, on reflection, that his answer has failed to capture the essence of entanglement. Any stream of particles, if not specially prepared, will measure +h/2 or -h/2 at detector A; they will do with respect to the x axis, or the y axis, or any axis. Exactly the same is true at detector B. How can this very ordinary circumstance illustrate the mystery of entanglement? But that seems to be what David has said: that if you prepare the particles in the entangled state, you get this "strange result". I see nothing strange about it since it seems to be exactly the same result if you set up two detectors far apart and measured streams of particles that were totally random.

I am going to suggest that the mystery of entanglement lies in the perfect correlation (or anti-correlation) that you get when you set up both detectors along the x axis. Some people think there is nothing mysterious about this because it is exactly what you would expect if the two particles were created with equal and opposite spins. These people are very wrong. The reason they are wrong is that the experiment works the same no matter how you align the detectors with respect to the source of the particles. We can imagine an experiment where particles are created with opposite spins, but assuming the spin axis is random at the moment of creation, there is no way a pair of detectors should show 100% correlation no matter what angle you set it to. In fact, in the case of entanglement, there is 100% correlation regardless of the orientation. That is a real problem and it is really the only problem.

EDIT: I explain this issue in more detail in my blog article, "Entanglement and the Crossed Polarizers".

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Actually, I don't agree. My answer refers to particles prepared in the eigenstate $|s_y = +\frac{\hbar}{2}\rangle$ (the positive eigenstate of y-spin), not particles taken from a sample with incoherent spin. The strange result, as described in my answer, is that a particle prepared with spin $s_y = +\hbar/2$ can be measured to have spin $s_y = -\hbar/2$ without going through an intermediate detector. Of course this is not the explanation you'd give to someone asking what entanglement is, but that's not quite the focus of the question. – David Z Jul 28 '11 at 2:15
OK, then I think you must be wrong. You seem to be saying that entangled particles are prepared with spin oriented in the y direction. Instead of |+-> - |-+> (singlet state)you are analyzing the case of |+-> + |-+> (triplet state)? Is that right? – Marty Green Jul 28 '11 at 3:07
No, I'm talking about a state $|\chi\rangle$ for which $\langle\chi|S_{y1}|\chi\rangle = \langle\chi|S_{y2}|\chi\rangle = +\frac{\hbar}{2}$, which neither of the $m_s = 0$ states you named satisfies. Although, to be fair, I haven't been working through the math behind the scenes. I'll see if I can do that and figure out exactly what state I am talking about, but it'll have to wait for a while as I have more urgent things to deal with tonight. – David Z Jul 28 '11 at 4:21
It just occurred to me, I should mention that I deleted my answer unless/until I can make a more mathematically precise statement. – David Z Aug 2 '11 at 20:54
I don't think this is a good answer. It could be the case that at the creation of the particle the outcome of measuring it at at an angle alpha is determined, for every such angle. What I mean is that a particle does not contain a single vector of polarization information, but rather it contains a full 360 degree set of predetermined outcomes. In that case you would also have perfect correlation. Bell's inequality is more subtle than this, because it excludes this possibility. – Jules Apr 20 at 10:00

There are no problems with second detector measurements. They occur as they would without first detector. The "problem" is that if you check them with the knowledge of the results of first detector, you might notice that measurements on both ends are correlated. If you measured x components in both, you definitely got opposite results. While if you check x in one and y in another you have no correlation at all. Which may be confusing.

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In a certain way, to see the paradox in the EPR experiment you have to have absorbed quantum mechanics into your blood. Otherwise one might not see the surprise in the result. I think that a much better paradox, for someone just learning the theory, is the GHZ experiment.

You begin with three photons in a linear superposition of two pure states. In the first pure state, all the photons are horizontally polarized HHH. In the second pure state, they are all vertically polarized VVV. In this situation, measuring any one photons horizontal versus vertical polarization immediately determines the measurements for the other two.

Something interesting happens when you make a measurement at an angle. It's too long to describe here, but it is elementary and by spending a few hours reading it and working it out you will see some amazing things. My feeling is that this is more of a surprise than the EPR experiment as it is an apparent contradiction in how we are used to using probabilities. In short, quantum mechanics seems to be about waves (and therefore complex numbers which encode the phase and amplitude of a wave) more than it is just about probabilities. It makes me sort of feel that our extraction of probabilities (i.e. the Born rule) is more accidental than fundamental; or that an answer to Einstein's complaint about QM is that the dice is our playing with the world (or perhaps vice versa) rather than that God does play with dice.

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It's not a "paradox". Einstein was troubled about the objective reality of complimentary variables. Before EPR, it was thought it's not possible to measure complimentarity variables simultaneously. He argued that some property has an objective value if without in any way disturbing it, we can know what it is with certainty. If we measure the z spin of an entangled pair, we automatically know if a measurement of the other z spin is made, it will have the opposite value. By his definition, the z spin of the other particle has now an objective existence. However, we may choose to measure the x spin of the first particle instead and the z spin of the other particle. The assumption he snuck in was counterfactual definiteness. Because we're no longer measuring the z spin of the first particle, we can't argue anymore that the z spin of the other particle has an element of reality prior to measurement. His argument falls flat if there's no counterfactual definiteness.

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As reading the wikipedia article, we can clearly see a way to transfer information faster than the speed of light, therefore we know there is an error in the article.

Let's say entangled particle pair has identical x-spins. We measure y-spin of one particle. The x-spin of the measured particle becomes random, the x-spin of the other particle is not affected.

Bohr debunked Einstein's, Podolsky's and Rosen's claim that, by using entangled particles, it is possible to measure both momentum and position accurately, by saying that measuring position causes the laboratory to have somewhat uncertain momentum, which cause the momentum measurement in the laboratory to be uncertain.

So don't you think Bohr's way to solve the EPR-paradox is the right way to solve the EPR-paradox?

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No, there is no way to actually transfer any information using only the entangled states. You also need some other communication channel to tell what was the actual measure done on the first particle. The speed of this signal is limited by the speed of light. This is explained well in the Wikipedia article. – Olof Jul 28 '11 at 9:30
@Olof: Here Wikipedia erroneously says that Alice's measurement turns Bob's measurement into lottery: 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 Bob's positron has a 50% probability of producing +x and a 50% probability of -x—so the outcome is 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 uncertain. – jartza Jul 29 '11 at 10:32
No, Let's assume we start with the entangled state $|++\rangle + |--\rangle$, where $\pm$ indicate spin in the $z$ direction. If Alice measures $s_z$ she gets $s_z = \pm 1/2$, with each outcome having a 50% probability. Then Bob measure $s_z$. He will always get the same result as Alice. If A instead measures $s_x$, and B later measures $s_z$ he will again get $s_z=\pm 1/2$ with 50% probability. So no matter what measurement Alice did, Bob will see the same result. To transfer any information, A will have to send a signal about what measurement she did. This signal is limited by $c$. – Olof Jul 29 '11 at 11:43
So isn't the quote from Wikipedia clearly incorrect? By the way, I have not said that you can transfer information instantaneously using entangled particles. – jartza Jul 29 '11 at 17:50

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