Editing Quantum operation (section)

==Quantum measurement==
Quantum operations can be used to describe the process of [[quantum measurement]]. The presentation below describes measurement in terms of self-adjoint projections on a separable complex Hilbert space ''H'', that is, in terms of a PVM ([[Projection-valued measure]]). In the general case, measurements can be made using non-orthogonal operators, via the notions of [[POVM]].  The non-orthogonal case is interesting, as it can improve the overall efficiency of the [[quantum instrument]].

=== Binary measurements ===
Quantum systems may be measured by applying a series of ''yes&ndash;no questions''. This set of questions can be understood to be chosen from an [[orthocomplemented lattice]] ''Q'' of propositions in [[quantum logic]]. The lattice is equivalent to the space of self-adjoint projections on a separable complex Hilbert space ''H''.

Consider a system in some state ''S'', with the goal of determining whether it has some property ''E'', where ''E'' is an element of the lattice of quantum ''yes-no'' questions. Measurement, in this context, means submitting the system to some procedure to determine whether the state satisfies the property. The reference to system state, in this discussion, can be given an [[operational definition|operational meaning]] by considering a [[statistical ensemble]] of systems. Each measurement yields some definite value 0 or 1; moreover application of the measurement process to the ensemble results in a predictable change of the statistical state. This transformation of the statistical state is given by the quantum operation
<math display="block"> S \mapsto E S E + (I - E) S (I - E). </math>
Here ''E'' can be understood to be a [[projection operator]].

===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]].