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 $H(p)+{\sum }_x{p}_{x}S({\rho }_x)$.

The classical-quantum state is block diagonal. If ${\lambda }_{xj}$ are the eigenvalues of ${\rho }_x$, its eigenvalues are $p_x{\lambda }_{xj}$. Inserting these into the definition of Von Neumann entropy gives the displayed sum of classical Shannon entropy and mean conditional quantum entropy. Its quantum mutual information is therefore the Holevo quantity $S({\sum }_x{p}_x{\rho }_x)-{\sum }_x{p}_{x}S({\rho }_x)$.

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.

Attach an orthogonal classical label register to an ensemble. Its quantum mutual information with the quantum system equals the ensemble Holevo quantity. A local quantum channel changes the ensemble states but preserves their labels and probabilities. Data processing for quantum mutual information then proves that the output Holevo quantity cannot exceed the input value.

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.

For $k$ equiprobable quantum code states with average decoding error $ϵ<1$, data-processing inequality for quantum relative entropy applied to the binary test for quantum decoding success givesfor $k\ge 2$. Here $h_2$ is binary entropy. The case $k=1$ is trivial. Taking the supremum over allowed input distributions yields a code-size bound in terms of the largest Holevo quantity; at $ϵ=1$ the undivided inequality is vacuous.