Editing Quantum operation (section)

== Definition ==

Recall that a [[density operator]] is a non-negative operator on a [[Hilbert space]] with unit trace.

Mathematically, a quantum operation is a [[linear map]] Φ between spaces of [[trace class]] operators on Hilbert spaces ''H'' and ''G'' such that
* If ''S'' is a density operator, Tr(Φ(''S'')) ≤ 1.
* Φ is [[Choi's theorem on completely positive maps|completely positive]], that is for any natural number ''n'', and any square matrix of size ''n'' whose entries are trace-class operators <math display="block"> \begin{bmatrix} S_{11} & \cdots & S_{1 n}\\ \vdots & \ddots & \vdots \\ S_{n 1} & \cdots & S_{n n}\end{bmatrix} </math> and which is non-negative, then <math display="block"> \begin{bmatrix} \Phi(S_{11}) & \cdots & \Phi(S_{1 n})\\ \vdots & \ddots & \vdots \\ \Phi(S_{n 1})  & \cdots & \Phi(S_{n n})\end{bmatrix} </math> is also non-negative. In other words, Φ is completely positive if <math>\Phi \otimes I_n</math> is positive for all ''n'', where <math>I_n</math> denotes the identity map on the [[C*-algebra]] of <math>n \times n</math> matrices.

Note that, by the first condition, quantum operations may not preserve the normalization property of statistical ensembles.  In probabilistic terms, quantum operations may be [[sub-Markovian]]. In order that a quantum operation preserve the set of density matrices, we need the additional assumption that it is trace-preserving.

In the context of [[quantum information]], the quantum operations defined here, i.e. completely positive maps that do not increase the trace, are also called [[quantum channel]]s or ''stochastic maps''. The formulation here is confined to channels between quantum states; however, it can be extended to include classical states as well, therefore allowing quantum and classical information to be handled simultaneously.