Editing Quantum channel (section)

=== Heisenberg picture ===

Density matrices acting on ''H<sub>A</sub>'' only constitute a proper subset of the operators on ''H<sub>A</sub>'' and same can be said for system ''B''.  However, once a linear map <math> \Phi</math> between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend <math> \Phi</math> uniquely to the full space of operators. This leads to the adjoint map <math> \Phi^*</math>, which describes the action of <math> \Phi</math> in the [[Heisenberg picture]]:{{sfn|Wilde|2017|at=§4.4.5}}

The spaces of operators ''L''(''H''<sub>''A''</sub>) and ''L''(''H''<sub>''B''</sub>) are Hilbert spaces with the [[Hilbert–Schmidt operator|Hilbert–Schmidt]] inner product. Therefore, viewing <math>\Phi : L(H_A) \rightarrow L(H_B)</math> as a map between Hilbert spaces, we obtain its adjoint <math> \Phi</math><sup>*</sup> given by

:<math>\langle A , \Phi(\rho) \rangle = \langle \Phi^*(A) , \rho \rangle .</math>

While <math> \Phi</math> takes states on ''A'' to those on ''B'', <math> \Phi^*</math> maps observables on system ''B'' to observables on ''A''. This relationship is same as that between the Schrödinger and Heisenberg descriptions of dynamics. The measurement statistics remain unchanged whether the observables are considered fixed while the states undergo operation or vice versa.

It can be directly checked that if <math> \Phi</math> is assumed to be trace preserving, <math> \Phi^*</math> is [[unital map|unital]], that is,<math> \Phi^*(I) = I</math>. Physically speaking, this means that, in the Heisenberg picture, the trivial observable remains trivial after applying the channel.