Exponential averaged spectral projectors for a finite cq ensemble

For $P_n$, retain product eigenvectors of ${\bar{\rho }}^{\otimes n}$ whose positive eigenvalue has negative logarithm in $n[S(\bar{\rho })-\eta ,S(\bar{\rho })+\eta ]$. For $Q_{x^n}$, use product eigenvectors of ${\rho }_{x^n}={⨂}_i{\rho }_{x_i}$ whose eigenvalue has negative logarithm in $n[s-\eta ,s+\eta ]$. The upper eigenvalue bounds are immediate. The lower retained eigenvalue and trace at most one imply each rank bound. Under the joint law $\mathrm{Pr}(X=x,J=j)=p_x{\lambda }_{xj}$, with zero-probability eigenvalues omitted, the random variable $Z=-\ln {\lambda }_{XJ}$ has finite support and mean $s$. By [F7], the negative logarithm of each product eigenvalue is the sum of the coordinate self-information variables, so its IID sum controls the second failure probability exactly. The first failure is the same argument for the spectral law of $\bar{\rho }$; averaging inputs gives ${\bar{\rho }}^{\otimes n}$.

For completeness, a finite supported centered variable $Y$ with $\mathrm{\mid }Y\mathrm{\mid }\le b$ obeys $\mathbb{E}{e}^{uY}\le 1+u^2{b}^2{e}^{\mathrm{\mid }u\mathrm{\mid }b}\mathrm{/}2$, by the Taylor remainder and $\mathbb{E}Y=0$. For small fixed $u>0$, $\ln \mathbb{E}{e}^{uY}<u\eta \mathrm{/}2$. Markov's inequality and independence then bound $\mathrm{Pr}(\sum Y_i\ge n\eta )\le e^{-nu\eta \mathrm{/}2}$; the logarithm product law [F7] gives the factor $n$ when taking the log of the product moment generating function. Apply the same argument to $-Y$ and add. A constant spectral variable has zero failure probability and needs no division by a range. All logarithms are evaluated only on positive-probability eigenvalues.