Convexity of the operational rates and regularized purification formula

Let $A,B$ be finite, nonempty types, and let $S$ consist of a vector $\psi\in\mathbb C^{A\times A}$ with $\|\psi\|=1$ and a positive semidefinite matrix $\Delta$ on $A\times B$. Write $\rho_\psi=|\psi\rangle\langle\psi|$, where $(|v\rangle\langle v|)_{ij}=v_i\overline{v_j}$, and write $H(X)=-\operatorname{Re}\operatorname{tr}[f(X)]$, with $f(x)=x\log_2x$ applied by continuous functional calculus. A channel here means a map $X\mapsto\sum_jK_jXK_j^\dagger$ with finitely many matrices satisfying $\sum_jK_j^\dagger K_j=I$; a density matrix means a positive semidefinite matrix of trace one. For a matrix $\omega$ on $A\times B$, let

where this infimum and all subsequent costs take values in $[0,\infty]$, and the set whose membership is asserted below is understood as $\{\max\{x,0\}:x\in\mathcal P(\omega)\}$. For a channel $N$ from $A^k$ to $B^k$, put $\omega_{k,N}=(\mathrm{id}_{A^k}\otimes N)(\rho_\psi^{\otimes k})$, with tensor factors grouped into $A^k\times B^k$, and for $k\ge1$ put

Here each $\omega_{A_iB_i}$ is obtained by tracing out every other letter. Define $R_a(D)$ as the infimum of all nonnegative real rates $r$ with the following property: for every $\epsilon>0$ and $s>0$, there is $n_0\in\mathbb N$ such that every $n\ge n_0$ with $n\ge1$ admits positive integer dimensions $q,c,h$, a shared density matrix on $\mathbb C^h\otimes\mathbb C^h$, and channel encoders $A^n\otimes\mathbb C^h\to\mathbb C^c\otimes\mathbb C^q$ and decoders $\mathbb C^c\otimes\mathbb C^q\otimes\mathbb C^h\to B^n$, such that the encoder output has zero entries between distinct classical labels for every input matrix, $\log_2q\le n(r+s)$, and the decoded output on $\rho_\psi^{\otimes n}$, leaving the reference registers untouched, has average distortion at most $D+\epsilon$. The assistance choice $a$ is arbitrary among unassisted ($c=h=1$), entanglement assisted ($c=1$), and classical assisted ($h=1$); no further dimension bound is imposed by this definition. For every $D_1,D_2\ge0$ and $0\le\lambda\le1$, the declaration asserts

It also asserts that, for every $D\ge0$, $F_{k+1}(D)$ converges in $[0,\infty]$ to $F_\infty(D)$, that $F_\infty(D)=R_{\mathrm{unassisted}}(D)$, and that for every integer $k\ge1$, if any channel $N:A^k\to B^k$ satisfies $d_k(\omega_{k,N})\le D$, then such a channel exists whose $E_P(\omega_{k,N})$ is no larger than that of every other channel satisfying the same distortion constraint. Finally, for every density matrix $\omega$ on $A\times B$, the infimum defining $E_P(\omega)$ belongs to the set of nonnegative entropy values defining it, so it is attained by a normalized purification with finite, positive ancillary dimensions. Empty feasible sets in the extended nonnegative infima have value $+\infty$; the convexity inequalities include $\lambda=0,1$, using $0\cdot(+\infty)=0$.