In quantum information theory, the classical capacity of a quantum channel is the maximum rate at which classical data can be sent over it error-free in the limit of many uses of the channel.

A mixed quantum state is a unit trace, positive operator known as a density operator, and is often denoted by ${\displaystyle \rho }$, ${\displaystyle \sigma }$, ${\displaystyle \omega }$, etc. The simplest model for a quantum channel is a classical-quantum channel

which sends the classical letter ${\displaystyle x}$ at the transmitting end to a quantum state ${\displaystyle \rho _{x}}$ at the receiving end, with noise possibly introduced in between. The receiver's task is to perform a measurement to determine the input of the sender. If the states ${\displaystyle \rho _{x}}$ are perfectly distinguishable from one another (i.e., if they have orthogonal supports such that ${\displaystyle \operatorname {Tr} \rho _{x}\rho _{x^{\prime }}=0}$ for ${\displaystyle x\neq x^{\prime }}$) and the channel is noiseless, then perfect decoding is trivially possible. If the states ${\displaystyle \rho _{x}}$ all commute with each other then the channel is effectively classical. The situation becomes nontrivial only when the states ${\displaystyle \rho _{x}}$ have overlapping support and do not necessarily commute.

The most general way to describe a quantum measurement is with a positive operator-valued measure, whose elements are typically denoted as ${\displaystyle \left\{\Lambda _{m}\right\}_{m}}$. These operators should satisfy positivity and completeness in order to form a valid POVM:

The probabilistic interpretation of quantum mechanics states that if someone measures a quantum state ${\displaystyle \rho }$ using a measurement device corresponding to the POVM ${\displaystyle \left\{\Lambda _{m}\right\}}$, then the probability ${\displaystyle p\left(m\right)}$ for obtaining outcome ${\displaystyle m}$ is equal to

and the post-measurement state is

if the person measuring obtains outcome ${\displaystyle m}$.

The above is sufficient to consider a classical classical communication scheme over a cq channel. The sender uses a cq channel to map a classical letter x to a quantum state ${\displaystyle \rho _{x}}$, which is then sent through some noisy quantum channel, and then measured using some POVM by the receiver, who obtains another classical letter.