Exact flat spectral-type reduction of channel information

Write $\rho ={\sum }_{j=1}^d{\lambda }_j\mathrm{\mid }j⟩⟨j\mathrm{\mid }$, including zero eigenvalues. For each count vector $t=(n_1,\dots ,n_d)$ with ${\sum }_j{n}_j=n$, let $m_t$ be the number of product basis words of that type, $p_t=m_t{\prod }_{j:n_j>0}{\lambda }_j^{n_j}$ (zero if any such ${\lambda }_j=0$), and omit types with $p_t=0$. Let $P_t$ project onto the span of the words of type $t$ and set ${\tau }_t=P_t\mathrm{/}{m}_t$. These states are flat, their supports are mutually orthogonal, and ${\rho }^{\otimes n}={\sum }_{t:p_t>0}{p}_t{\tau }_t\mathrm{.}$ There are at most $(n+1{)}^d$ types. If $N_T$ is their number and $u$ is the uniform law on them, [F6] gives $0\le D(p\mathrm{\parallel }u)=\ln N_T-H(T)$, so $H(T)\le d\ln (n+1)$. Since $S({\tau }_t)=\ln m_t$, the logarithm product law [F5] and ${\sum }_t{p}_t{n}_j=n{\lambda }_j$ yield ${\sum }_t{p}_{t}S({\tau }_t)={\sum }_t{p}_t\ln m_t=nS(\rho )-H(T)\mathrm{.}$

Let $B_t={\mathcal{N}}^{\otimes n}({\tau }_t)$ and $E_t=({\mathcal{N}}^c{)}^{\otimes n}({\tau }_t)$, with averages $\bar{B}={\sum }_t{p}_t{B}_t=\mathcal{N}(\rho {)}^{\otimes n}$ and $\bar{E}={\sum }_t{p}_t{E}_t={\mathcal{N}}^c(\rho {)}^{\otimes n}$. By [F4], the receiver-minus-environment entropy identity applies to each type. The upper entropy-excess bound [F1] gives ${\sum }_t{p}_{t}S(B_t)\ge S(\bar{B})-H(T)$, while its lower entropy-excess bound gives ${\sum }_t{p}_{t}S(E_t)\le S(\bar{E})$. Product spectral decompositions and [F5] give $S(\bar{B})-S(\bar{E})=n[S(\mathcal{N}(\rho ))-S({\mathcal{N}}^c(\rho ))]=n{I}_c(\rho ,\mathcal{N})$. Hence ${\sum }_t{p}_t{I}_c({\tau }_t,{\mathcal{N}}^{\otimes n})\ge n{I}_c(\rho ,\mathcal{N})-H(T)\mathrm{.}$ Adding ${\sum }_t{p}_{t}S({\tau }_t)=nS(\rho )-H(T)$ from step 1.1 gives the second inequality.

Each displayed left side in step 2.1 is an average over the finite positive-probability types, so at least one type attains at least the corresponding average lower bound. These are exact finite-$n$ estimates; no replacement of fixed-dimensional continuity by a uniform bound is involved.