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?