Classical-quantum state (CQ state)

A classical-quantum state has block-diagonal form ${\rho }_{XB}={\sum }_x{p}_x|x⟩⟨x|\otimes {\rho }_x$. Its entropy is $H(p)+{\sum }_x{p}_{x}S({\rho }_x)$.

For an ensemble $\{p_x,{\rho }_x\}$ with average $\bar{\rho }$, the Holevo quantity is $\chi =S(\bar{\rho })-{\sum }_x{p}_{x}S({\rho }_x)=I(X:B)$ of the associated classical-quantum state.

The classical mutual information obtained by any measurement of a quantum ensemble is at most its Holevo quantity. This follows from the data-processing inequality for quantum relative entropy applied to the measurement channel.