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Alice measures the spin of her electron on the x axis. She now knows the spin value of Bob's electron on the x axis at time T0. Bob measures the spin of his electron on the z axis. He now knows the spin value of Alice's electron on the z axis at time T0. The two meet up and speak, knowing both the x and z values of their electrons at T0, contradicting the uncertainty principle, which states that an electron "doesn't have" both the values at the same time.

There is obviously something wrong in what I wrote. What's that?

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    $\begingroup$ There are a lot of problems, starting from the fact that the original EPR-paper exactly establishes the fact that there are particles where measuring one part of the system determines the other part without choosing the measuring axis in advance to the fact that the uncertainty relation is a statistical property (I can very well first measure X and then Z and will obtain two outcomes). $\endgroup$ – Martin Sep 23 '15 at 22:20
  • $\begingroup$ But if you do that it's useless, cause the act of measuring would change the state. I'm referring to this statement from wikipedia: "In quantum mechanics, the x-spin and z-spin are "incompatible observables", meaning the Heisenberg uncertainty principle applies to alternating measurements of them: a quantum state cannot possess a definite value for both of these variables." $\endgroup$ – user2502368 Sep 23 '15 at 22:25
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Under your assumption of simultaneously well-defined x and z value, you reach predictions which are inconsistent with quantum theory. This is exactly what leads to Bell's inequality which is (experimentally!) violated by quantum theory, see https://en.wikipedia.org/wiki/Bell%27s_theorem or the explanation in Preskill's lecture notes (http://www.theory.caltech.edu/~preskill/ph229/notes/chap4_01.pdf, Section 4.2).

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  • $\begingroup$ Thanks for the links! So, what would be the result (referring to the state at T0)? Does the act of measuring the x axis on one electron influence the sign of the spin on the z axis of the other electron? It wasn't my intention to assume, I wanted to know what the result would represent (I understand that the data cannot represent the state of the electron at T0, I'm not sure why) $\endgroup$ – user2502368 Sep 23 '15 at 22:41
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    $\begingroup$ Wait, maybe I got it. When Bob measures the z axis, he's just measuring his electron, and can get no valuable information about Alice's, because Alice's electron doesn't have a defined z value? $\endgroup$ – user2502368 Sep 23 '15 at 22:50
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    $\begingroup$ @user2502368 Exactly. And if Alice and Bob assume that their outcomes allow them to infer the outcome of an unperformed measurement in the same basis, they get a correlation inequality -- Bell's inequality -- which is violated by quantum mechanics. (As Asher Peres put it: "Unperformed experiments have no results.") $\endgroup$ – Norbert Schuch Sep 23 '15 at 23:02
  • $\begingroup$ Doesn't Bell's inequality (Aspect's experiments) only contradict the locality principle, leaving the possibility of non-local variables? In Bohm's theory, "there is always a matter of fact about the position and momentum of a particle. Each particle has a well-defined trajectory. Observers have limited knowledge as to what this trajectory is [...]". I'm just saying things to understand if they are correct, please don't think I'm correcting you or something. I'm very sorry. $\endgroup$ – user2502368 Sep 23 '15 at 23:06
  • $\begingroup$ If you assume local realism (i.e., observables have an predetermined outcome independent of whether they are measured, and there is a notion of a light-cone), then this implies to Bell's inequality, which is violated by experiments. Therefore, theories which are both local and realistic are ruled out. $\endgroup$ – Norbert Schuch Sep 24 '15 at 6:14
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Start with two particles in the state $UD+DU$, where $U$ and $D$ are the "up" and "down" eigenstates of the measurement Alice plans to make.

Put $u=U+D$ and $d=U-D$, and suppose Bob makes a measurement with these eigenstates. Note that $UD+DU=ud+du$. (The factor of 2 doesn't matter).

Alice makes her measurement and finds her electron is in state, say, $U$. This puts the pair of electrons in the state $UD=U(u+d)$. Bob makes his measurement and has a 50-50 chance of finding his electron in state $u$ or $d$, putting the pair in state $Uu$ or $Ud$.

Or alternatively, Bob measures in the same direction Alice did, in which case he finds his electron in state $D$ and the pair remains in state $UD$.

For Alice to believe (or to know) that the pair is in state $UD$, she must believe (or know) that Bob either a) made no measurement or b) measured in the same direction that she did. Otherwise, all she knows is that the pair is in the state $Ux$, where $x$ could be anything.

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