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Suppose I have a pair of entangled particles, each one goes in a different direction. Now I measure the momentum of one particle and the position of the other at the same time. Because the measurements are space-like, there's a frame of reference in which one occurs before the other, and vice versa. Thus, in one frame of reference we end up with a momentum eigenfunction and in the other with a position one - since the momentum and position operators do not commute. How does this make sense with the formalism of QM or QFT? Apparently the 'collapse of wavefunction' interpretation is not well-defined.

Thanks in advance for the answers, Eran

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  • $\begingroup$ You don't end up with a momentum eigenfunction when you take the measurement, your measurement device is put in a macroscopic state that was correlated with the momentum eigenvalue of the particle when it was measured. You don't know what state the particle is in after the measurement, you do know it is impossible for the particle to in a momentum eigenstate though, that's not an allowable element of the Hilbert space. $\endgroup$
    – Daniel Kerr
    Commented Jul 11, 2016 at 21:08
  • $\begingroup$ If I understand it correctly, those ideas are more inherent to QM, much like the uncertainty principle. Can't we assume a perfect measurement? But even with imperfect measurements you apparently end up with two different wavefunctions that in each case are correlated to another state. $\endgroup$
    – eranreches
    Commented Jul 11, 2016 at 21:25
  • $\begingroup$ A perfect measurement would not disturb the state, right? So then there would be no collapse, the state out would be the same as the state in. Is that what you mean by a "perfect" measurement? A measurement is necessarily an entanglement with a measurement device, yes. $\endgroup$
    – Daniel Kerr
    Commented Jul 11, 2016 at 21:36
  • $\begingroup$ Collapse was never a useful concept. The only question of interest is... why/how do useless concepts hang around for so long? $\endgroup$
    – CuriousOne
    Commented Jul 12, 2016 at 5:05

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It does not matter which measurement happens first, and there is no privileged frame: in both frames, after the corresponding measurements, the quantum state of one particle evolves from a sharp position, while the other particle evolves from a state with a sharp momentum.

This is what "wavefunction collapse" means in that case: measuring either momentum or position breaks the corresponding entanglement. There will be no more correlation between the particles positions nor momenta.

For a similar situation with other observables, let's suppose both particles are entangled with opposite spins. Then if you measure spin $A$ along $z$ and spin $B$ along $x$ you will not find any correlation, but now the spin entanglement is broken (in whatever frame, whatever measurement comes first): if you later measure spins $A$ and $B$ along $z$, you will find no correlation.

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  • $\begingroup$ Thanks for the answer! Still something bothers me. From my understanding the first measurement doesn't break the entanglement, so suppose one measure the momenta of one particle first. After the measurement we get a system wavefunction corresponding to a joint state of the particles in which the first has the specific momenta we've measure. Now the probabilities for measuring the position of the second differ from the situation where the measurements are in reverse order. So there might be a position with high probability in one order and lower in another. Which prediction wins? $\endgroup$
    – eranreches
    Commented Jul 12, 2016 at 7:23
  • $\begingroup$ As you said in the question, there is no first measurement per se, since which measurement happens first depends on the frame. A measurement is a boundary condition of the quantum state; in that sense, a single measurement does break the entanglement. $\endgroup$
    – Stéphane Rollandin
    Commented Jul 12, 2016 at 12:09
  • $\begingroup$ I agree we can't determine which measurement happened first, but it seems to me that there's no self-consistency i.e. the predictions we make depend on the frame of reference. Even if we treat the measurements as boundary conditions. $\endgroup$
    – eranreches
    Commented Jul 12, 2016 at 17:29

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