Editing Quantum channel (section)

=== Classical information ===

So far we have only defined a quantum channel that transmits only quantum information. As stated in the introduction, the input and output of a channel can include classical information as well. To describe this, the formulation given so far needs to be generalized somewhat. A purely quantum channel, in the Heisenberg picture, is a linear map Ψ between spaces of operators:

:<math>\Psi : L(H_B) \rightarrow L(H_A)</math>

that is unital and completely positive ('''CP'''). The operator spaces can be viewed as finite-dimensional [[C*-algebra]]s. Therefore, we can say a channel is a unital CP map between C*-algebras:

:<math>\Psi : \mathcal{B} \rightarrow \mathcal{A}.</math>

Classical information can then be included in this formulation. The observables of a classical system can be assumed to be a commutative C*-algebra, i.e. the space of continuous functions <math>C(X)</math> on some set <math>X</math>. We assume <math>X</math> is finite so <math>C(X)</math> can be identified with the ''n''-dimensional Euclidean space <math>\mathbb{R}^n</math> with entry-wise multiplication.

Therefore, in the Heisenberg picture, if the classical information is part of, say, the input, we would define <math>\mathcal{B}</math> to include the relevant classical observables. An example of this would be a channel

:<math>\Psi : L(H_B) \otimes C(X) \rightarrow L(H_A).</math>

Notice <math>L(H_B) \otimes C(X)</math> is still a C*-algebra. An element <math>a</math> of a C*-algebra <math>\mathcal{A}</math> is called positive if <math>a = x^{*} x</math> for some <math>x</math>. Positivity of a map is defined accordingly.  This characterization is not universally accepted; the [[quantum instrument]] is sometimes given as the generalized mathematical framework for conveying both quantum and classical information. In axiomatizations of quantum mechanics, the classical information is carried in a [[Frobenius algebra]] or [[Frobenius category]].