Editing Quantum operation (section)

===General case===
In the general case, measurements are made on observables taking on more than two values.

When an observable ''A'' has a [[Self-adjoint operator#Pure point spectrum|pure point spectrum]], it can be written in terms of an [[orthonormal]] basis of eigenvectors.  That is, ''A'' has a spectral decomposition
<math display="block"> A = \sum_\lambda \lambda \operatorname{E}_A(\lambda)</math>
where E<sub>''A''</sub>(λ) is a family of pairwise orthogonal [[Orthographic projection|projection]]s, each onto the respective eigenspace of ''A'' associated with the measurement value λ.

Measurement of the observable ''A'' yields an eigenvalue of ''A''. Repeated measurements, made on a [[statistical ensemble]] ''S'' of systems, results in a probability distribution over the eigenvalue spectrum of ''A''.  It is a [[discrete probability distribution]], and is given by
<math display="block"> \operatorname{Pr}(\lambda) = \operatorname{Tr}(S  \operatorname{E}_A(\lambda)).</math>

Measurement of the statistical state ''S'' is given by the map
<math display="block"> S \mapsto \sum_\lambda \operatorname{E}_A(\lambda) S \operatorname{E}_A(\lambda)\ .</math>
That is, immediately after measurement, the statistical state is a classical distribution over the eigenspaces associated with the possible values λ of the observable: ''S'' is a [[mixed state (physics)|mixed state]].