heisenberg uncertainty principle - Simultaneously measurement in quantum mechanics?

In quantum mechanics $A$ and $B$ can be simultaneous measured if mathematically $\hat{A}\hat{B}=\hat{B}\hat{A}$. But how do we actually measure thing simultaneously. $\hat{A}\hat{B}$ is not simultaneous measurement because because here we measure $B$ and then $A$. So the above mathematical equation only tells us that $\hat{A}\hat{B}$ ($B$ first, $A$ later) = $\hat{B}\hat{A}$ ( $A$ first, $B$ later) - how does this relate to simultaneous measurement?

Answer

simultaneous measurement doesn't mean that commuting operators are measured simultaneously (in time, as you asked in question) and not one by one. It means that when the two operators commute, then they are diagonal in the same ket basis, which are referred to as simultaneous eigenkets, and you can now measure them one by one without disturbing the collapsed state vector. The fact that they are diagonal in the basis of same eigenkets allows them to be measured simultaneously without disturbing the state vector. Now coming to your example of two commuting operators, it does not matter in which order you measure the operators, since the two operators commute with each other. You can measure them like $ XY$ or $YX$.

And simultaneous measurement is not prohibited by quantum mechanics. It is prohibited for those operators that don't obey commutation relations and thus are not diagonal in the same ket basis, and therefore the state vector is disturbed each time they are measured. A famous example is that of position and momentum, they have a non-zero commutator and thus cannot be measured simultaneously.