Hello all, I've been thinking about the connection between commutativity of operators and uncertainty. I've convinced myself that to have simultaneous eigenstates is a necessary and sufficient condition for two observeables to be measured simultaneously and accurately. It's also clear that simultaneous eigenstates gives us commutativity. So we have that non-commuting observables have an uncertainty relation between them. What's not exactly clear to me is why the converse holds, ie commuting observeables can be measured simultaneously and accurately. Is there simple proof of this that I'm missing?