Finite classical-quantum states and Holevo information

Let $X$ be a finite nonempty classical label set, represented by a register $H_X$ with computational orthonormal basis $({\xi }_x{)}_{x\in X}$ (Finite registers and their ordered computational bases). For each $x$, write ${\mathrm{\Pi }}_{x}z:=⟨z,{\xi }_x⟩{\xi }_x$, the rank-one orthogonal projection using the root linear-first convention, as in An orthogonal finite ensemble block sum is a density operator. Let $H_B$ be a nonzero finite-dimensional complex inner-product space. A finite classical-quantum state is

${\rho }_{XB}:={\sum }_{x\in X}{p}_x{\mathrm{\Pi }}_x\otimes {\rho }_x,$

where $p_x\ge 0$, ${\sum }_x{p}_x=1$, and each ${\rho }_x$ is a density operator on $H_B$. It is a density operator by An orthogonal finite ensemble block sum is a density operator. Its quantum marginal is ${\bar{\rho }}_B:={\sum }_x{p}_x{\rho }_x={\mathrm{Tr}}_X{\rho }_{XB}$ by the first-factor partial-trace formula in Partial trace over the first tensor factor, and it is a density operator by Partial traces of a finite bipartite state are states. The Holevo information of this ensemble is

$\chi (\{p_x,{\rho }_x{\}}_{x\in X}):=S({\bar{\rho }}_B)-{\sum }_{x:p_x>0}{p}_{x}S({\rho }_x),$

where zero-probability branches contribute nothing. This is measured in nats under Finite-dimensional von Neumann entropy.