Why can't be the EPR experiment simplified? 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?
 A: 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).
A: 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.
