Commuting Observables and Compatibility
Observables can be measured simultaneously with definite values only if they commute; non-commuting observables obey an uncertainty relation.
When can two things be known at once
Two observables A and B are compatible — simultaneously measurable with definite values — if and only if they commute, meaning AB = BA. The commutator [A,B] = AB - BA measures the failure of commutation, and it governs which quantities can share a definite value.
The mathematical reason
Commuting Hermitian operators can be diagonalised in the same basis; they share a complete set of common eigenvectors. In such a shared eigenstate, both observables have definite values at once. If the operators do not commute they have no common eigenbasis, so no state gives both definite values.
- [A,B] = 0: shared eigenbasis, jointly measurable
- [A,B] != 0: no shared eigenbasis, an uncertainty trade-off applies
Qubit example
The Pauli operators do not commute: XZ = -ZX, so [X,Z] = 2XZ != 0. A qubit cannot have definite X and Z values simultaneously — this is why a state definite in the computational (Z) basis, like |0>, is a 50/50 superposition in the X basis. Measuring one scrambles the other.
Measurement disturbance
With non-commuting observables, measuring A disturbs B. Measure Z on |+> and you get a random 0 or 1; the qubit is now |0> or |1>, and a subsequent X measurement is now random. The order of incompatible measurements matters, unlike compatible ones. This is a direct operational consequence of non-commutation.
Connection to uncertainty
The size of the commutator sets a lower bound on the product of the spreads of A and B — the uncertainty principle. Commuting observables have no such bound and can both be sharp. Compatibility is therefore the dividing line between quantities that coexist and quantities that trade off.