Quantum channel

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.)

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 Φ between density matrices acting on ${\displaystyle H_{A}}$ and ${\displaystyle H_{B}}$ with the following properties:

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