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I’m having trouble understanding what exactly happens when one makes a measurement on one particle, especially since I am seeing conflicting information on here, on Wikipedia, and from other sources.

The top answer here by Luboš Motl implies that entanglement is just a correlation. But he also says,

The measurement of one particle in the pair "causes" the wave function to collapse, which "actively" influences the other particle, too. The first observer who measures the first particle manages to "collapse" the other particle, too. This picture is, of course, flawed. The wave function is not a real wave.

Meanwhile, the Wikipedia article states,

Now suppose Alice is an observer for system A, and Bob is an observer for system B. If in the entangled state given above Alice makes a measurement in {0,1} eigenbasis of A, there are two possible outcomes, occurring with equal probability:

  1. Alice measures 0, and the state of the system collapses to 0(A) 1(B)
  1. Alice measures 1, and the state of the system collapses to 1(A) 0(B)

If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1. If the latter occurs, (Alice measures 1) then Bob's measurement will return 0 with certainty. Thus, system B has been altered by Alice performing a local measurement on system A. This remains true even if the systems A and B are spatially separated. This is the foundation of the EPR paradox.

The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see no-communication theorem.

Although both of these statements seem to agree that no actual communication occurs between one particle to the other, the wiki seems to suggest that the joint wave function’s state is realized, whereas the stack exchange answer seems to suggest that there is only a knowledge update. Which is it?

Most importantly, before measurement, is it the case that the joint wave function is described differently (where it is indefinite) compared to after the measurement where it results in a definite value?

Qmechanic
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2 Answers2

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What we can measure are just the results of the experiments and the probabilities that result from many measurements of the same realization, not the wave function.

As you are interested in the math and note the interpretations I will take that route. According to the standard formulation of quantum mechanics (as in Postulates of quantum mechanics) you have to use the Born rule. The state is given by $$|\Psi\rangle=\frac1{\sqrt2}(|01\rangle+|10\rangle)\tag{1}$$ and after Alice measures, the state is either $$|\Psi\rangle=|01\rangle\tag{2a}$$ or $$|\Psi\rangle=|10\rangle\tag{2b}$$

The joint state is written differently before and after measurement. Standard quantum mechanics does not tell you how this happens. You have two options:

  • Agnostic but human-centered view: $|\Psi\rangle$ is a math trick that may or not simulate what actually happens. The calculation is carried by somebody, in this case by Alice. According to standard QM, Alice needs write differently the joint wave function before and after she makes the measurement.

  • Stronger but non-anthropocentric view: The joint wave function a real thing out there that collapses instantaneously after Alice's measurement.

Mauricio
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Nobody can actually say when the joint (entangled) wave function changes to a product state. Experiments clearly show that it makes no more sense to say that Alice's measurement before Bob changes Bob any more than Bob's later measurement changes the earlier Alice outcome. It's only an assumption either way.

How do we know that? Well, it's a prediction of QM that it is the joint context - and nothing else - that describes the resulting statistics. And... you can entangle particles after they have been detected. Time ordering makes no difference under any scenario (as to the observed outcomes).

Consequently, there might be an effect when one thing is done before another. But nothing like that has ever been observed in tests. So: it is only after the ENTIRE experimental context is realized that you can be sure you are truly in a post-measurement "collapsed" state (assuming you call it that).

DrChinese
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