quantum mechanics - Sending Information With Entangled Particles

I learned from my quantum mechanics course that if you measure a quantum state twice, two things can happen:

1) You take the second measurement just after the first on. In this case, the result will be the same. The wave function has not yet "de-collapsed", so to say.

2) You wait a little before you take the second measurement. This time, the two measurements are not correlated, and the second result is again random.

Now relate this to the EPR-paradox. If you have two entangled particles far away from each other, and you do a measurement on one of the particles, you don't know if you got the result as a consequence of someone measuring the other particle or if it was simply random. This is the argument used, when someone tries to explain why no information is sent by entangled particles (and thus, they don't disagree with relativity or violate causality).

But what if those who measure the first particle were to continually do measurements? If you then try to do measurements (in intervals long enough that the second case described above should occur) on the second particle, you will keep getting the same result as a consequence of those measuring the first particle. Thus, it is possible to conclude whether or not those in the first lab are doing this kind of continuous measurements right now or not. Consequentially, information has been sent potentially with speed faster than light.

Answer

Mark Mitchison’s comment is an excellent basis for understanding what entanglement is.

Entanglement is a type of correlation between the random outcomes of local measurements on quantum particles.

In other words

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  It is possible to arrange a correlation for two quantum particles
  A correlation for two particles could be based on spin or the orientation of their electrical or magnetic dipole (polarization) or electrical charge, etc..... For example, a free neutron decays into an electron and a proton (and an anti-neutrino). It's a spontaneous process and you can't know which particle is moving in which direction. But when you measure the electric charge of one of these particles, you know the charge of the other particle. The particles are correlated.

  
  


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  The measurement shows a random outcome of the measured parameter
  In the case of neutron decay, the result is binary. You are able to measure the electric field of an electron or proton and the result is unique. If one measures the electron on one side, then one undoubtedly measures the proton on the other side.
  Not so for photons. The only available parameter beside their energy content is the orientation of their electric field component. Their orientation is randomly distributed over 360°. The measurement of polarization is done by a polarization filter, which let through all particles say from zero to 90° and from 180° to 270°.

  
  


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  Entanglement is a type of correlation An uncertainty in the measurement makes such a correlation interesting (and at the beginning pleasantly mysterious). For neutron decay, a trapped electron can be measured several times and the electron still has the same intrinsic electric field. And you are able to send another electron to a receiver and the manipulation will not be remarkable for the receiver (despite a time delay that you can compensate beforehand).
  Not so for photons. In the ideal case you are able to get a coincidence of 50% for the measurements. in the other cases the photon simply does not reach the detector.
  Furthermore any influence chances the direction of polarization (more correct, the direction of the field components). The measurement is not repeatable.

  
  

I learned ... that if you measure a quantum state twice, two things can happen:
1) You take the second measurement just after the first on. In this case, the result will be the same. The wave function has not yet "de-collapsed", so to say.
2) You wait a little before you take the second measurement. This time, the two measurements are not correlated, and the second result is again random.

I'm not sure you understood what you learned. As long as the particles are undisturbed, the correlation will be the same independent from the time delay of the measurement. The undisturbance in real life is the problem, not the time delay.

It’s worth to answer the next paragraphs of your question as long as the miss-concept at the beginning is not corrected.