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In the paper "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete ?", it implies that with an entangled state it is possible to know two physical quantities simultaneously. For example lets say Alice and Bob is entangled with a singlet state of spin. Then I measure z axis of Alice, that would tell me the definite value of Bob's spin in z axis. But if Alice changes her measurement to x axis that would also tell me the definite value of Bob's spin in x axis. And therefore, Alice knows Bob's z,x axis spin value simultaneously.

Here's my question, when Alice measures her spin in the z axis then that measurement will collapse the singlet sate into a pure tensor product. Meaning that it's no longer entangled. So, there is no way that Alice can then measure x axis and know Bob's spin value.

Where am I wrong?

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If you accept:

  1. locality (Alice's measurement does not disturb Bob's particle and Bob's measurement does not disturb Alice's particle)
  2. independence (Alice's choice of what to measure does not imply a change at Bob's location)

it follows that both spins (on X or Z) must be real. The idea is not that Alice measures the x-spin and then she measures the Z spin, but the fact that Alice could, in principle, measure either X or Z spin.

The locality assumption is difficult to deny and was accepted by Bohr as well. The second assumption is not so obvious and it offered Bohr an opportunity to attack the argument. Einstein himself was not happy with the paper either.

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Here's my question, when Alice measures her spin in the z axis then that measurement will collapse the singlet sate into a pure tensor product. Meaning that it's no longer entangled. So, there is no way that Alice can then measure x axis and know Bob's spin value.

Indeed you are right. What you are missing is that this is done in separate experimental runs.

You do a bunch of experiments with Alice and Bob measuring on the $z$ axis. Then you do some other experiments with Alice and Bob using some other measurement axis, and so on.

At each experimental run, a new (but identical to the previous ones) singlet state is given to Alice and Bob to measure.

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