In quantum information theory, a quantum channel is a communication channel that can transmit quantum information, as well as classical information. An example of quantum information is the general dynamics of a qubit. An example of classical information is a text document transmitted over the Internet.

Terminologically, 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 include trace-decreasing maps while reserving "quantum channel" for strictly trace-preserving maps.)

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.

Consider quantum channels that transmit only quantum information. This is precisely a quantum operation, whose properties we now summarize.

Let ${\displaystyle H_{A}}$ and ${\displaystyle H_{B}}$ be the state spaces (finite-dimensional Hilbert spaces) of the sending and receiving ends, respectively, of a channel. ${\displaystyle L(H_{A})}$ will denote the family of operators on ${\displaystyle H_{A}.}$ In the Schrödinger picture, a purely quantum channel is a map ${\displaystyle \Phi }$ between density matrices acting on ${\displaystyle H_{A}}$ and ${\displaystyle H_{B}}$ with the following properties:

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 ${\displaystyle \Phi }$ is only required to be not trace-increasing. In this article, it will be assumed that all channels are CPTP.

Density matrices acting on H_A only constitute a proper subset of the operators on H_A and same can be said for system B. However, once a linear map ${\displaystyle \Phi }$ between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend ${\displaystyle \Phi }$ uniquely to the full space of operators. This leads to the adjoint map ${\displaystyle \Phi ^{*}}$, which describes the action of ${\displaystyle \Phi }$ in the Heisenberg picture: