In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of commuting observables whose simultaneous eigenspaces span the Hilbert space, so that the eigenvectors are uniquely specified by the corresponding sets of eigenvalues. Since each pair of observables in the set commutes, the observables are all compatible so that the measurement of one observable has no effect on the result of measuring another observable in the set. It is therefore not necessary to specify the order in which the different observables are measured. Measurement of the complete set of observables constitutes a complete measurement, in the sense that it projects the quantum state of the system onto a unique and known vector in the basis defined by the set of operators. That is, to prepare the completely specified state, we have to take any state arbitrarily, and then perform a succession of measurements corresponding to all the observables in the set, until it becomes a uniquely specified vector in the Hilbert space (up to a phase). The compatibility theorem Consider two observables, and , represented by the operators and . Then the following statements are equivalent: and are compatible observables. and have a common eigenbasis. The operators and commute, meaning that . Proofs Discussion We consider the two above observables and . Suppose there exists a complete set of kets whose every element is simultaneously an eigenket of and . Then we say that and are compatible. If we denote the eigenvalues of and corresponding to respectively by and , we can write If the system happens to be in one of the eigenstates, say, , then both and can be simultaneously measured to any arbitrary level of precision, and we will get the results and respectively. This idea can be extended to more than two observables. Examples of compatible observables The Cartesian components of the position operator are , and . These components are all compatible. Similarly, the Cartesian components of the momentum operator , that is , and are also compatible. Formal definition A set of observables is called a CSCO if: All the observables commute in pairs. If we specify the eigenvalues of all the operators in the CSCO, we identify a unique eigenvector