Eq.(44): equivalence of AEP and operational cost identity

For finite index types $A,B$, with decidable equality, let $N:A\to B$ be a channel, meaning a finite Kraus sum with $\sum_iK_i^\dagger K_i=I_A$. The declaration asserts an equivalence between the iterated-limit statement and the operational equality defined here. Put $S(X)=-\operatorname{Re}\operatorname{tr}\operatorname{cfc}(x\mapsto x\log_2x)(X)$, $v_\rho(a,a')=(\sqrt\rho)_{a,a'}$, $R_\rho=(\mathrm{id}_A\otimes N)(|v_\rho\rangle\langle v_\rho|)$, and

Density matrices are positive semidefinite with trace $1$. Define $D_{\max}(R\|T)=\inf\{\lambda\in\mathbb R:2^\lambda T-R\ge0\}$ in the extended reals, $I_{\max}(R)=\inf_{\sigma\text{ density}}D_{\max}(R\|(\operatorname{Tr}_B R)\otimes\sigma)$, and

where $\Phi$ is the normalized maximally entangled matrix on the input and an identical reference, with entry the inverse input dimension when both index pairs have equal coordinates and $0$ otherwise; either extended-real infinity converts to $0$. The diamond norm uses all matrices of trace norm at most $1$ on an input-sized reference tensor the input, with $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$. The iterated-limit statement is that there exists $L:\mathbb R\to\mathbb R$ such that, for every $0<e<1$,

To define the operational side, for $F:U\to V$ and $G:X\to Y$, let $E_o(F,G)$ be the infimum of $\tfrac12\|\mathcal W_F(P)-G\|_\diamond$, where

Nonsignalling assistance permits channels $P:X\times V\to U\times Y$ whose two output marginals, on every matrix, are channel functions of their respective input marginals. Entanglement assistance permits precisely maps obtained by adjoining a density matrix on two finite positive-dimensional ancillary systems and applying a channel locally to each input together with its ancillary share. Let $Q_{\mathrm{ent}}(N)$ be the extended-real supremum of nonnegative real $r$ such that, for every $e>0$ and slack $s>0$, eventually every positive $n$ admits a positive integer $m$ with $E_{\mathrm{ent}}(N^{\otimes n},\mathrm{id}_m)\le e$ and $(r-s)n\le\log_2m$. Let $C_{\mathrm{ns}}(N)$ be the extended-real infimum of nonnegative real $r$ with the corresponding conditions $E_{\mathrm{ns}}(\mathrm{id}_m,N^{\otimes n})\le e$ and $\log_2m\le(r+s)n$. Tensor powers act by $(N^{\otimes n}(X))_{y,z}=\sum_{x,x'}X_{x,x'}\prod_iN(|x_i\rangle\langle x'_i|)_{y_i,z_i}$. The asserted equivalence is that the stated iterated limit holds if and only if $Q_{\mathrm{ent}}(N)=C_{\mathrm{ns}}(N)$; the operational error conditions concern infima and do not assert realizing assistance maps.