Measuring two components at once using Entanglement Heisenberg's uncertainty principle states that it is impossible to measure two properties of a particle (like $S_z$ and $S_x$) with certainty at the same time. Consider the following experiment (from this page):

The middle part is a radioactive substance (of total spin of zero) which emits electron pairs in opposite directions. The right filter is oriented at 0 degrees (measuring $S_z$) and the left one oriented at 90 degrees (measuring $S_x$). 
Here is the part which I don't understand: if an electron passes through the left filter, it means that $S_x$ = +h/2. Therefore, the corresponding right electron has $S_x$ = -h/2. Now, the right electron is also passing through the $S_z$ filter at the same time, and hence we can measure $S_z$. 
Isnt this against the uncertainty principle and what does the principles of relativity have to say about this? Sure it will take time to communicate the $S_x$ result, but we are measuring $S_x$ and $S_z$ at the same time. Therefore, aren't we able to measure both of the properties at the same time, even though the result of one measurement is communicated at a later time?   
 A: If you observe the left-hand electron to get through the filter, this is equivalent to preparing the right-hand electron in the state $S_x=-\hbar/2$. Actually, you could have done that without entanglement. Just take an unpolarized electron beam that's going to the right, and pass it through a filter that requires $S_x=-\hbar/2$. Then pass it through a second filter that requires $S_z=+\hbar/2$. This doesn't mean that the electron has definite values of both $S_x$ and $S_z$ simultaneously. After the $S_z$ measurement, the electron no longer has a definite value of $S_x$.
BTW, there is no uncertainty principle of the standard form for $S_x$ and $S_z$, although they are not compatible observables, just like $p$ and $x$ are not compatible observables. But you can make this paradox into an equivalent one by having the electrons be emitted from a particle at rest in the lab frame, so they're in entangled states $p_1=-p_2$. Then you can put them through $x$ and $p$ filters. The resolution of the paradox is the same.
A: This seems to correspond to the Einstein-Podolsky-Rosen (EPR) paradox (in the Bohm formulation) which has since been resolved.
