Based on my knowledge of quantum entanglement, I can set up a scenario which leads to a contradiction with the No-communication theorem. Please help me find the flaw.

Suppose Alice wants to communicate a message to Bob (1 or 0). Alice has 100 electrons (which we will call a_1, a_2, ..., a_100) and Bob also has 100 electrons (which we will call b_1, b_2, ..., b_100). Each one of Alice's electrons is entangled to exactly one of Bob's electrons such that a_n and b_n have opposite spin. First, Alice measures the spin of each of her electrons along the z axis and records each result. Then, if Bob wants to communicate a zero, he does not measure the spin of his electron, so that when Alice measures her's a second time about the z axis, she gets the same results as in her first measurement. If Bob wants to communicate a 1, he measures the spin of each one of his electrons about the x axis so that when Alice makes a second measurement of each of her electrons about the z axis, her results should not be correlated to the first measurement she took. If everything I said were true, this would provide a very high probability for Bob to correctly transfer a bit of information to Alice, potentially at faster than light speeds which violates the No-communication theorem.

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    $\begingroup$ once Alice has measured her electrons the wavefunction collapses and the entanglement is broken, so Bob's measurements (or non-measurements) do not affect Alice's electrons anymore $\endgroup$
    – glS
    Feb 23 '15 at 21:30

Once Alice makes the measurement, her state is collapsed and her spin will not be affected by Bob's subsequent measurement. Bob's spin is now determined in the z-direction, however, which is what allows the usual quantum key-distribution techniques.

To be more precise:

Before Alice's measurement the state is $$|+ \rangle |-\rangle - |-\rangle |+\rangle.$$

After, if Alice measures a positive spin, it is $$|+\rangle|-\rangle.$$

Then, Bob measures, but this only changes his spin: $$|+\rangle (|-\rangle + |+\rangle)$$

Alice's spin is not acted on.

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    $\begingroup$ Ok, but then it seems to me that entanglement just refers to our knowledge of how parts the a system relate to each other. How do we know that the particles' spins are not just corollated, but that the particles are actually in communication with each other? $\endgroup$
    – Sheepman
    Feb 23 '15 at 21:47
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    $\begingroup$ That's a fascinating question, and ultimately I don't think I can give you an answer correctly. This forms the core of the "EPR paradox." Einstein believed there was a "hidden variable theory" (the spins are already determined even if we don't know them) that would prevent this famous "spooky action at a distance." The counter-argument is quite technical, but it can be shown that the correlations are unlike any correlations that would be seen in a hidden variable theory. This was proven in "Bell's theorem," although off the top of my head I can't think of any good nontechnical explanations. $\endgroup$
    – zeldredge
    Feb 23 '15 at 21:53
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    $\begingroup$ @SamuelMontgomery: We do not know that the particles are "in communication" with each other because they are not in communication with each other. On the other hand, entanglement is certainly not just correlation, because the statistics of Alice and Bob's measurements violates Bell's Theorem. $\endgroup$
    – WillO
    Feb 23 '15 at 23:18

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