 A quantum state is a specification, for every experiment that can be performed on the system,  of probabilities for the possible outcomes of that experiment. These can be summed up as an assignment of an expectation value to each observable. These states are required to be linear. This means that, if an operator \(C\), corresponding to some observable, is the sum of operators \(A\) and \(B\), corresponding to other observables, then the expectation value that a quantum state assigns to \(C\) must be the sum of the expectation values assigned to \(A\) and \(B\). This is a nontrivial constraint, as it is required to hold whether or not the observables represented are compatible. A quantum state, therefore, relates expectation values for quantities yielded by incompatible experiments.