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Concerning quantum entanglement Wikipedia says:

Measurements of physical properties such as position, momentum, spin, and polarization, performed on entangled particles are found to be correlated. For example, if a pair of particles is generated in such a way that their total spin is known to be zero, and one particle is found to have clockwise spin on a certain axis, the spin of the other particle, measured on the same axis, will be found to be counterclockwise, as is to be expected due to their entanglement. However, this behavior gives rise to seemingly paradoxical effects: any measurement of a property of a particle performs an irreversible collapse on that particle and will change the original quantum state. In the case of entangled particles, such a measurement will be on the entangled system as a whole.

I am worried about the phrase "measured on the same axis". If one particle is measured on the earth on a certain axis and the other is measure on the moon (or somewhere near Alpha Centauri) how does one set up the measuring devices so that both are measured on the "same axis"? It seems this would require some kind of parallel transport which might be difficult or impossible to do--especially if curvature of space-time must be taken into account.

And what if the two measuring locations are a billion or so light years apart. Must we wait for a settled theorem of quantum gravity to deal with this case?

I imagine something like a Stern-Gerlach apparatus as measuring device.

In the EPR experiments done on the surface of the earth, what devices are used and how are they aligned.

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  • $\begingroup$ Even if the measurements are made on the earth!If a measurement is made on the north pole and the other at the equator, what is the axis? An axis going from the north to the south pole? An "moving" axis parallel to the gravitational field? In fact, as I also have been thinking in the same question for a while, I partially read some papers like arxiv.org/abs/quant-ph/0610030, arxiv.org/abs/1111.1864 and arxiv.org/abs/1111.1853, but no one of them answers exactly what I wanted. I would really appreciate if someone could answer your question. $\endgroup$ – jobe May 21 at 20:32
  • $\begingroup$ Also to complicate matters, I understand that the measurements don't have to be made simultaneously. So what to do if you wait a year or so to perform the second measurement? $\endgroup$ – W. Edwin Clark May 21 at 22:07
  • $\begingroup$ Excellent question. But at the scales you mentioned, you do not have to go quantum, even if you try to measure classical conservation of momentum along an axis, you will likely face similar challenges. $\endgroup$ – kpv May 23 at 0:40
  • $\begingroup$ @jobe In case you're still interested, I posted an answer. $\endgroup$ – Chiral Anomaly May 24 at 4:55
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I don't think the question was about quantum correlation, but about the meaning of "in the same direction" at remote locations in curved spacetime. Operationally, you would point the two detectors at the same remote galaxy and that would be good enough FAPP. But theoretically in a curved spacetime you should parallel transport the direction of detector A to the location of detector B. Then the problem appears that the result will depend on the path chosen for the parallel transport. Different paths from A to B will produce different directions at B proportional to the curvature and the area enclosed by the two paths. So we may expect that the exact anti-correlation of measured spins will be spoiled.

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This is simpler than it sounds.

The state space for a given particle is (a quotient of) a two-dimensional vector space. We can pick any two orthogonal vectors we want to, call them $U$ and $D$, and use them as a basis for this space. Then we call a particle in state $U$ "spin up", a particle in state $D$ "spin down", a particle in state $U+D$ anything we want (say "spin left"), a particle in state $U-D$ anything else we want (say "spin right"), etc.

If you have two particles, you have two state-spaces. Pick a basis $U_1,D_1$ for the first and a basis $U_2,D_2$ for the second. The choice of these bases establishes a (perfectly arbitrary) isomorphism between the state spaces. Identifying them along this isomorphism, we can write $U$ for either $U_1$ or $U_2$ and $D$ for either $D_1$ or $D_2$.

Now if the first particle is in state $U_1$ and the second in state $U_2$, we describe their spin states as "the same". If we'd chosen a different isomorphism, we'd desccribe these states as "different". But nothing hinges on this choice of words.

After all, if I start with the basis $U_1,D_1$ for my particle and you start with the basis $U_2,D_2$ for yours, then a pair in the entangled state $U_1U_2+D_1D_2$ will always be found in "the same" spin state according to the conventions above. If you change your mind and use the basis $D_2,U_2$ (changing the order), then the very same particles will always be found in "opposite" spin states --- but the meaning of "opposite" has changed, so we're still saying the same thing in different words.

Likewise if you change your basis to, say, $U_1-D_1,U_2+D_2$. Now the correct statement becomes more complicated: If I measure in what I call the "up" direction and you measure in what you call the "right" direction, then our observations agree.

TL;DR: We can each arbitrarily label any direction we want Up and the opposite direction Down. Our choices will change the way we describe the outcomes of experiments, but won't actually change those outcomes.

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