system
Given the following post, respond as if you're commenting on Reddit:
human
Why must two quantum mechanical observables have their commutator equal zero if they are to be simultaneously observable?
gpt
If the operators commute, they can be simultaneously diagonalized (assuming they have non-degenerate spectra for simplicity). That means that they share a basis of eigenstates. If you have two operators A and B with [A,B] = 0, then you can express any state in your Hilbert space as a linear combination of states of the form |a,b>, where A|a,b> = a|a,b>, and B|a,b> = b|a,b>.

If you start with the state |a,b> and measure the value of observable B, you get b, and the system remains in the state |a,b>. Then if you subsequently measure A, you will get a, and the system will still stay in the state |a,b>. So you have simultaneously measured A and B; the state of the system never changed, and you know the value of both of these observables.