In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information is a text document transmitted over the Internet. More formally, quantum channels are completely positive (CP) trace-preserving maps between spaces of operators. In other words, a quantum channel is just a quantum operation viewed not merely as the reduced dynamics of a system but as a pipeline intended to carry quantum information. (Some authors use the term "quantum operation" to also include trace-decreasing maps while reserving "quantum channel" for strictly trace-preserving maps.) Memoryless quantum channel We will assume for the moment that all state spaces of the systems considered, classical or quantum, are finite-dimensional. The memoryless in the section title carries the same meaning as in classical information theory: the output of a channel at a given time depends only upon the corresponding input and not any previous ones. Schrödinger picture Consider quantum channels that transmit only quantum information. This is precisely a quantum operation, whose properties we now summarize. Let and be the state spaces (finite-dimensional Hilbert spaces) of the sending and receiving ends, respectively, of a channel. will denote the family of operators on In the Schrödinger picture, a purely quantum channel is a map between density matrices acting on and with the following properties: As required by postulates of quantum mechanics, needs to be linear. Since density matrices are positive, must preserve the cone of positive elements. In other words, is a positive map. If an ancilla of arbitrary finite dimension n is coupled to the system, then the induced map where In is the identity map on the ancilla, must also be positive. Therefore, it is required that is positive for all n. Such maps are called completely positive. Density matrices are specified to have trace 1, so has to preserve the trace. The adjectives completely positive and trace preserving used to describe a map are sometimes abbreviated CPTP. In the literature, sometimes the fourth property is weakened so that is only required to be not trace-increasing. In this article, it will be assumed that all channels are CPTP. Heisenberg picture Density matrices acting on HA only constitute a proper subset of the operators on HA and same can be said for system B. However, once a linear map