§5 unnumbered extension: approximately recoverable channels

Fix finite nonempty index sets $A,B$, with decidable equality. There exist real $C_q,C_p\ge0$ and $0<\delta\le2$, chosen independently of the environments below, such that the following holds for every pair of finite nonempty index sets $E,F$, also with decidable equality, and every pair of complex matrices $V:A\to B\times E$, $W:B\to A\times F$ satisfying $V^\dagger V=I_A$ and $W^\dagger W=I_B$. Write

where receiver and complementary maps always trace out respectively the environment and receiver registers. First, for every complex matrix $\rho$ on $A$, with no density assumption, $\operatorname{Tr}_{F}(N_U^c(\rho))=N_V^c(\rho)$. Second, for every $\varepsilon\in[0,2]$,

Here $\|T\|_\diamond=\sup_{\|X\|_1\le1}\|(\operatorname{id}_{H}\otimes T)(X)\|_1$ for input index set $H$, with arbitrary complex $X$ on $H\times H$, and $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. Third, whenever $0\le\varepsilon\le\delta$ and the same recovery error is at most $\varepsilon$, put $e=\operatorname{rank}J(N_V)$, where $J(N_V)_{(a,b),(a',b')}=N_V(|a\rangle\langle a'|)_{bb'}$, and define

The matrix function uses spectral functional calculus, $\log_2x=\log x/\log2$, and $0\log_2 0=0$. Let $Q(V)$ be the supremum of nonnegative rates $r$ admitting, for every $n\ge0$, positive message dimensions $M_n$ and encoding and decoding channels through $N_V^{\otimes n}$, whose entanglement transmission fidelities tend to $1$ and whose rates satisfy, for every $s>0$, eventually $r-s\le\log_2M_n/n$. This fidelity is the overlap with $|\Phi_{M_n}\rangle\langle\Phi_{M_n}|$ after the encoded channel and decoder act on the second half of $|\Phi_{M_n}\rangle=M_n^{-1/2}\sum_m|m,m\rangle$. Let $P(V)$ be the supremum of nonnegative rates $r$ such that, for every $\epsilon',s>0$, all sufficiently large positive $n$ admit $M>0$, input density matrices $\rho_m$, and positive semidefinite decoding matrices $F_j$ summing to $I$, with $r-s\le\log_2M/n$ and

All channels in these definitions are finite Kraus sums with $\sum_jK_j^\dagger K_j=I$, and density matrices are positive semidefinite with trace $1$. If $e=1$ or $\varepsilon=0$, the conclusion is exactly $Q(V)=I_c(V)$ and $P(V)=I_c(V)$. Otherwise the conclusion is

Both error ranges include $\varepsilon=0$; only the capacity assertions use the smaller range $[0,\delta]$. The quantum code sequence includes $n=0$, with real division by zero interpreted as $0$.