My Particularly Troublesome Version of the EPR Paradox

[Simultaneity]

In quantum physics, a wave function instantaneously collapses once an observable is measured. For instance, if an electron's angular momentum in the z direction is measured, then the angular momenta in the x and y direction immediately become indeterminate due to the Heisenberg Uncertainty Principle. If two particles are entangled and described by a single wave function, such as two particles that are products of a previous particle that decayed and whose combined spin is known due to conservation of angular momentum, and an observable is measured on one of them, the wave function instantaneously collapses on the other, no matter how far apart the particles are. Experiments can and have been done to show instantaneous transmission of information (such as which direction the measurement took place along). But special relativity says that there is no such thing as a universal "instant" for all observers in all frames of reference. So, suppose Observer A is watching a spaceship pass by at a very fast constant velocity. Observer B is in the middle of the ship. Two light beams are emitted, one from the front of the ship (the edge of the ship farthest along the direction of motion according to Observer A), such that Observer B observes they have been simultaneously emitted at the same instant. Observer A sees that the light beam from the back of the ship takes longer to reach the middle than the light beam from the front, because it must catch up with the ship, while the ship is racing to meet the light beam emitted from the front. Both observers agree on the speed of light, which is universal. But if the light beams reach Observer B at the same time according to Observer B, Observer A must also see this: If Observer B sets up a machine that kills a cat if both beams reach him at the same time, Observer A would obviously agree that the cat is dead (physics thought experiments are a dangerous place for cats). So Observer A must see the beam of light being emitted from the back of the spaceship before the beam of light is emitted from the front of the spaceship in order for the beams to meet in the middle. If a series of clocks is placed along the length of the ship that Observer B sees as synchronized, Observer A will see the clocks that are further forward in the direction of motion as reading fractions of a second less, depending on how far forward they are. So, now we take our two particles that are entangled, and we put one at the front of the ship and one at the back of the ship. Two machines are set up that measure each particle's spins, and the machines each take a measurement along a different direction. Observer B sees the particle at the front of the ship being measured slightly before the particle at the back of the ship, while Observer A sees the opposite, because the back of the ship is further along in time for Observer A than it is for Observer B. So what happens to the wave function? Which axis does it collapse along? It seems natural that Observer B might be fundamentally correct, since he isn't moving with respect to the particles, but suppose we have four particles that are entangled, and the other two are in corresponding positions on the ship that Observer A is on. Observer B sees himself as being still and Observer A as moving, so neither observer is still with respect to the particles. This is not an extremely technical paradox: The mathematics only require an undergraduate level of physics education, but to my knowledge no one has ever resolved this paradox.

 

[Dale]

Hi Simultaneity, welcome to PF!

Can you please use paragraphs to make your post more readable, and edit your post for brevity?

Although it is not a direct response to your question, it seems that you may be unaware that modern QFT is fully Lorentz invariant, meaning that any modern quantum mechanics respects special relativity.

 

[lugita15]

So, now we take our two particles that are entangled, and we put one at the front of the ship and one at the back of the ship. Two machines are set up that measure each particle's spins, and the machines each take a measurement along a different direction. Observer B sees the particle at the front of the ship being measured slightly before the particle at the back of the ship, while Observer A sees the opposite, because the back of the ship is further along in time for Observer A than it is for Observer B. So what happens to the wave function? Which axis does it collapse along? It seems natural that Observer B might be fundamentally correct, since he isn't moving with respect to the particles, but suppose we have four particles that are entangled, and the other two are in corresponding positions on the ship that Observer A is on. Observer B sees himself as being still and Observer A as moving, so neither observer is still with respect to the particles.

It is indeed true that observers in different frames will disagree about which measurement broke the entanglement (i.e. collapsed the wavefunction of the two-particle system). And according to the principle of relativity, you can't really say that either one of them is "fundamentally correct" and the other one is "fundamentally incorrect". Each of them is correct with respect to their own reference frame, just like different frames will disagree as to what the length or speed of an object is.

But the important thing is that this leads to no observable difference. Suppose you have a pair of entangled photons, which are sent to different detectors which measure their polarization. Each photon either goes through the detector or does not go through depending on whether it is detected as being polarized parallel or perpendicular to the orientation of the detector. Then regardless of the orientation of the two detectors, it doesn't matter which detector first detects one of the photons. When either one of the photons reaches the detector, it will seem to go through or not go through with 50-50 probability. So just based on looking at what one photon does, you can't tell whether entanglement has been broken yet or not. It's only when you compare the results of the two photons that you notice the correlation, and that correlation is the same regardless of which photon went through the detector first. (This means, by the way, that you can't use entanglement to send messages faster than the speed of light, even though the wavefunction collapse occurs instantaneously; detecting the entanglement requires comparing the results of the two detectors, and that can only be done by slower-than-light means, like going from the location of one detector to that of the other, or by using ordinary communication to talk to someone at the other detector.)

By the way, if you want to learn more about EPR and Bell's theorem, I suggest that you read http://quantumtantra.com/bell2.html, which I think contains the simplest known proof of Bell's theorem.

 

[lugita15]

Although it is not a direct response to your question, it seems that you may be unaware that modern QFT is fully Lorentz invariant, meaning that any modern quantum mechanics respects special relativity.

There is certainly a sense in which quantum field theory is more "relativity-friendly" than nonrelativistic quantum mechanics; nonrelativistic QM is based on the Galilei group, whereas modern QFT is based on the Poincare group. But in another sense, both of them are compatible with relativity. You can say that each of them is as local and as nonlocal as the other. They are both nonlocal in the sense that wavefunction collapse is instantaneous, even if two entangled particles are spacelike-seperated. (Whether this is a "real" phenomenon is of course a hotly debated interpretational question.) But they both respect the no-communication theorem, which states quantum phenomena cannot make information propagation occur faster than it would otherwise be able to occur. So in reference to the OP, as long as FTL communication is impossible in the absence of quantum entanglement, you can't use quantum entanglement to achieve it.