Classical capacity

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. Holevo, Schumacher, and Westmoreland proved the following least upper bound on the classical capacity of any quantum channel ${\displaystyle {\mathcal {N}}}$:

${\displaystyle \chi ({\mathcal {N}})=\max _{\rho ^{XA}}I(X;B)_{{\mathcal {N}}(\rho )}}$

where ${\displaystyle \rho ^{XA}}$ is a classical-quantum state of the following form:

${\displaystyle \rho ^{XA}=\sum _{x}p_{X}(x)\vert x\rangle \langle x\vert ^{X}\otimes \rho _{x}^{A},}$

${\displaystyle p_{X}(x)}$ is a probability distribution, and each ${\displaystyle \rho _{x}^{A}}$ is a density operator that can be input to the channel ${\displaystyle {\mathcal {N}}}$.