Eq.(68): constant-channel divergence form of the AEP

For finite types $A,B$ with decidable equality and every channel $N:\operatorname{Mat}(A)\to\operatorname{Mat}(B)$, there exists a real number $v$ such that the infimum, over all density matrices $\sigma$ on $B$, of $D_{\mathrm{ch}}(N\|\mathcal R_\sigma)$ equals the finite extended-real value $v$, where $\mathcal R_\sigma(X)=\operatorname{tr}(X)\sigma$. A density matrix is positive semidefinite with trace $1$, and a channel has a finite Kraus representation whose Kraus matrices satisfy $\sum_jK_j^\dagger K_j=I$. Here $D_{\mathrm{ch}}(N\|M)$ is the supremum over all unit vectors $\psi\in\mathbb C^{A\times A}$ of the base-two relative entropy between $(\mathrm{id}_A\otimes N)(\psi\psi^\dagger)$ and $(\mathrm{id}_A\otimes M)(\psi\psi^\dagger)$; this relative entropy is $+\infty$ unless both arguments are density matrices and $\ker\sigma\subseteq\ker\rho$, and otherwise is the nonnegative part of $\operatorname{Re}\operatorname{tr}(\rho\log\rho-\rho\log\sigma)$, divided by $\log2$, with the matrix functions defined by continuous functional calculus and the total real logarithm. In addition, for every real $e\ge0$ and every natural $n$, including $n=0$, the following extended-real infimum $R_n(e)$ is neither $+\infty$ nor $-\infty$:

Here $\sigma$ is on $B^n$, $J_T((a,b),(a',b'))=T(|a\rangle\langle a'|)_{b,b'}$, tensor powers are defined by products of matrix-unit entries, and $\|F\|_\diamond=\sup_{\|X\|_1\le1}\|(\mathrm{id}_I\otimes F)(X)\|_1$ for input register $I$, using arbitrary matrices $X$ on $I\times I$, with $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$. Finally, there exists a function $L:\mathbb R\to\mathbb R$ such that, for every $0<e<1$, $R_{n+1}(e)/(n+1)\to L(e)$ as $n\to\infty$, and $L(e)\to v$ as $e\downarrow0$. This is an iterated limit; the assertion gives no uniformity in $e$. Extended-real infima and suprema retain their infinite values, and conversion to a real sends either infinity to $0$, although the stated finiteness excludes that conversion case for $R_n(e)$ when $e\ge0$.