I know how this can be proved using the quantum relative entropy. However, even with this proof, and am still confused about how this emerges.
Say I have a source that produces two states $\rho_1$ and $\rho_{2}$ with probability a half each, and both are mixed states, ie $S(\rho_i)>0$ for each of them. The dimensions of the Hilbert space is $2$.
How can $S(\rho)=H(p_{i})+\sum_{i}p_{i}S(\rho_{i})\le \log(d)$, given that $H(p_{i})=1$ and $S(\rho_i)>0$, given that $\log(d)=\log(2)=1?$
I am assuming that I am missing something obvious in the actual construction of $\rho$, in that something is bounding $H(p_{i})$ away from 1. I am assuming this has something to do with orthogonal supports, as $S(\rho) \le H(p)+\sum_{i}p_iS(\rho_{i})$ if they are not orthogonal.