The answers here about experimentally verifying Heisenberg's uncertainty principle says the following.
Step 3, select two operators A and B
Step 4a, for some of the systems prepared in state $\Psi$, measure A
Step 4b, for some of the systems prepared in state $\Psi$, measure B
Why not measure $A$ on all members of the ensemble followed by a measurement of B on each? What is the problem if the first measurement (say, A) collapses the state $\Psi$ to an eigenstate of A?
Addendum If this is the case i.e., A and B are measured on different members of the ensemble, there is apparently no correlation between the measurments. Then what does it mean to say as $\Delta A$ decreases by measuring A more and more accurately, $\Delta B$ increases when we measure B?
Does any textbook explain these steps of measurement?