Editing Quantum channel (section)

=== The cb-norm of a channel ===

Before giving the definition of channel capacity, the preliminary notion of the '''norm of complete boundedness''', or '''cb-norm''' of a channel needs to be discussed. When considering the capacity of a channel <math>\Phi</math>, we need to compare it with an "ideal channel" <math>\Lambda</math> . For instance, when the input and output algebras are identical, we can choose <math>\Lambda</math> to be the identity map. Such a comparison requires a [[metric (mathematics)|metric]] between channels.
Since a channel can be viewed as a linear operator, it is tempting to use the natural [[operator norm]]. In other words, the closeness of <math>\Phi</math> to the ideal channel <math>\Lambda</math> can be defined by

:<math>\| \Phi - \Lambda \| = \sup \{ \| (\Phi - \Lambda)(A)\|  \;|\;  \|A\| \leq 1 \}.</math>

However, the operator norm may increase when we tensor <math>\Phi</math> with the identity map on some ancilla.

To make the operator norm even a more undesirable candidate, the quantity

:<math>\| \Phi \otimes I_n \|</math>

may increase without bound as <math>n \rightarrow \infty.</math> The solution is to introduce, for any linear map <math>\Phi</math> between C*-algebras, the cb-norm

:<math>\| \Phi \|_{cb} = \sup _n \| \Phi \otimes I_n \|.</math>