No, you can't do this. Any process like this would make quantum mechanics incompatible with special relativity. Different inertial frames would disagree about the ordering, and predict different results. It would let you do experiments to determine an absolute reference frame for the universe.
More mathematically, you can prove that the matrix $A \otimes I$ commutes with the matrix $I \otimes B$. Combined with the deferred measurement principle, this shows the ordering of operations on the two parts of the system cannot matter.