Classical channel vs quantum channel

[aaaa202]

I have a hard time understanding what a channel that transmits classical information is in the framework of quantum information theory. My textbook says the following:

Let P(ylx) be a classical channel defined by a conditional probability distribution. We define the corresponding quantum channel by:

∑_x,y (Tr lx><xlX)P(ylx)ly><yl

How should I interpret this formula and what is the role of the trace appearing? And in general what is a classical channel as opposed to a quantum channel? I guess the answer is that a classical channel can send definite states given by some probability distribution, while quantum channels are maps between states in general.

Another place in the book it states that a classical channel of two bits is of the form:
C(X) = ∑_i,j lij><ijl Tr lij><ijlX
Is this equivalent with the first formula?

 

[naima]

Could you explain what is X in your question?
Have you links to this subject?

 

[naima]

The first thing which comes to my mind is that the trace of $\rho X$ is the mean value of X in the state $\rho$
Now if X is an operator it sends X to another operator. How can we describe it? What is the image of a projector |z><z|?