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?