Abstract main result: both low-noise capacities agree with one-shot coherent information to leading orders

For every finite nonempty type $A$ equipped with decidable equality, there exist real constants $C\ge0$ and $0<\delta\le1$ such that, for every finite nonempty type $E$ equipped with decidable equality, every complex matrix $V:A\times E\leftarrow A$ satisfying $V^\dagger V=I_A$, and every real $0<\varepsilon<\delta$, the assumption $\|N-\mathrm{id}\|_\diamond\le\varepsilon$ implies

Here $N(\rho)=\operatorname{Tr}_E(V\rho V^\dagger)$, $N^c(\rho)=\operatorname{Tr}_A(V\rho V^\dagger)$, and

using spectral functional calculus. For a map $\Phi$ with input index type $I$, $\|\Phi\|_\diamond$ denotes the real supremum of $\|(\mathrm{id}_I\otimes\Phi)(X)\|_1$ over all complex matrices $X$ on $I\times I$ with $\|X\|_1\le1$, where $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$, and amplification applies $\Phi$ separately to each reference matrix block. A channel means a map $\rho\mapsto\sum_jK_j\rho K_j^\dagger$ for a finite family with $\sum_jK_j^\dagger K_j=I$. The number $Q(N)$ is the real supremum of nonnegative rates $r$ admitting, for every natural block length $n$, a positive message dimension $M_n$, a channel encoding that dimension into $A^n$, and a channel decoding $A^n$ back into it, such that the overlap of $(\mathrm{id}\otimes(\mathrm{Dec}_n\circ N^{\otimes n}\circ\mathrm{Enc}_n))(|\phi_{M_n}\rangle\langle\phi_{M_n}|)$ with $|\phi_{M_n}\rangle\langle\phi_{M_n}|$ tends to one, and, for every $s>0$, eventually $r-s\le\log_2M_n/n$; here $\phi_M=M^{-1/2}\sum_{j=0}^{M-1}|j,j\rangle$. The number $P(V)$ is the real supremum of nonnegative rates $r$ such that, for every $\eta>0$ and $s>0$, some positive $n_0$ ensures that every $n\ge n_0$ admits a positive number $M$ of density matrices $\rho_m$ on $A^n$ and positive semidefinite measurement matrices $F_j$ on $A^n$ summing to the identity, with $r-s\le\log_2M/n$ and $\|\Omega-J_M(\sigma)\|_1<\eta$, where, writing $V_n((b_i),(e_i)),(a_i)=\prod_iV_{(b_i,e_i),a_i}$,

Tensor powers act on tensor products of matrix units by the corresponding tensor products of their images. The constants $C,\delta$ are chosen before $E,V,\varepsilon$, so the same constants apply to all those choices; $\varepsilon=0$ and $\varepsilon=\delta$ are excluded from the asserted bounds.