%===================================================================== % Em05-A.tex % % Standalone manuscript for pdfLaTeX. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error Em05-A.tex % pdflatex -interaction=nonstopmode -halt-on-error Em05-A.tex % % No external bibliography, figures, numerical data, source files, % shell escape, or program execution are needed for typesetting. % % EVIDENCE BOUNDARY % This source supplies analytical proofs. % Compilation, numerical execution, independent symbolic execution, % formal proof checking, and physical implementation are not asserted. % % PHYSICAL BOUNDARY % The extraction parameter is a channel parameter, not evaporation time. % Controlled extraction is not identified with passive evaporation. % No gravitational dictionary, Page curve, or black-hole microstate % identification is constructed. % % IMPORTANT CORRECTIONS % - A specified decoder is not presumed optimal. % - Perfect transfer is an endpoint of the declared finite channel % family, not an asserted finite-time continuum outcome. % - The dephasing experiment has an additional retained environment. % - Entanglement cuts and diamond-distance normalization are explicit. % - Bell-input subsystem spectra are not asserted for arbitrary inputs. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,tabularx,array} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\C}{\mathbb C} \newcommand{\R}{\mathbb R} \newcommand{\ii}{\mathrm i} \newcommand{\Id}{I} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle #1|} \newcommand{\kb}[2]{|#1\rangle\langle #2|} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\Hil}{\mathcal H} \newcommand{\NN}{\mathcal N} \newcommand{\CC}{\mathcal C} \newcommand{\DD}{\mathcal D} \newcommand{\RR}{\mathcal R} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\id}{\mathrm{id}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddiam}{d_\diamond} \newcommand{\ee}{\varepsilon} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\spec}{spec} \DeclareMathOperator{\diag}{diag} \title{\textbf{Em05-A}\\[0.5em] Excitation Transfer, Subsystem Purity,\\ and Referenced Information Recovery\\[0.4em] \large Exact Diagnostics and Population-Preserving Noise in a Fermionic Extraction Channel} \author{Ngo Cheung\\ Independent Researcher\\ Hong Kong SAR, China} \date{Version 1} \begin{document} \maketitle \begin{abstract} We analyze a completely specified even-parity excitation-qubit extraction channel and its full complementary channel. The signed isometry yields exact formulas for transferred occupation, occupation-basis coherence, subsystem purity, reference correlations, and deterministic recovery performance. For a Bell input, we determine the accessible and residual spectra, their mutual-information balance, and the reference--accessible negativity. A complete specified decoder has half-diamond error \(1-\eta\), Bell fidelity \((1+3\eta)/4\), worst-case target overlap \(\eta\), and decoded reference concurrence \(\eta\). We distinguish these formulas from optimal recovery and determine the optimal errors at zero and full transfer. An independently declared population-preserving output dephasing experiment retains its additional environment. At full transfer, its optimal recovery error is \((1-\lambda)/2\), while the Bell-input accessible marginal, occupation, and purity are independent of \(\lambda\). Complete dephasing therefore leaves perfect occupation delivery and an unchanged accessible marginal, but destroys reference entanglement and prevents exact quantum recovery. These results provide a self-contained information-flow benchmark. They neither identify the channel with black-hole evaporation nor establish a gravitational Page curve. No numerical execution or completed verification package is asserted. \end{abstract} \section{Scope, prior work, and the diagnostic question} \label{sec:scope} \subsection{The question} The following quantities are often considered together: \begin{enumerate} \item excitation transferred to an accessible subsystem; \item coherence in a specified occupation basis; \item entanglement across a specified partition; \item purity of an accessible or residual marginal; \item recoverability of an unknown quantum message, including its correlations with an external reference. \end{enumerate} They answer different questions. This manuscript constructs an exact comparison within a specified channel family. In particular, we prove that a pure accessible subsystem can contain no message information, and that a mixed accessible subsystem can carry a perfectly recoverable quantum message. We also construct two experiments with identical Bell-input accessible marginals and identical excitation populations, but different optimal quantum-recovery errors. \subsection{Established background and contribution boundary} Finite-dimensional evaporation models have substantial precedent. Avery studies a class encompassing unitary and nonunitary toy evaporation processes \cite{Avery}. Hayden and Preskill formulate an information-retrieval task under assumptions including unitary, rapidly mixing internal dynamics and extensive access to radiation \cite{HaydenPreskill}. Neither model is identified with the extractor specified below. Reference-based exact recovery is part of the established quantum-error-correction framework \cite{KnillLaflamme}. Channel continuity and information--disturbance relations are developed by Kretschmann, Schlingemann, and Werner \cite{KSW}. B\'eny and Oreshkov formulate approximate-correctability conditions and an environmental dual optimization using worst-case entanglement fidelity \cite{BenyOreshkov}. Our contribution is the explicit diagnostic comparison, complete signed channel construction, and population-preserving noise counterexample for the declared extraction family. We do not claim a new general information--disturbance principle, a new definition of quantum recovery, or first-in-literature priority for the elementary dephasing and replacement-channel calculations. The finite channel is restated and proved independently here. The supplied Em03-H-v8 and Em04-A-v2 manuscripts motivate the comparison, but are not formal premises of its proofs. The XY-chain diagnostic identities of Em04-A are not imported into this different excitation code. \subsection{Evidence and physical boundaries} Ordinary quantum states, channels, tensor products, and Born probabilities are assumed. The mathematical results are analytical. No compiler log, numerical archive, executed symbolic check, machine-checked proof, or physical receiver accompanies this source. The parameter \(\eta\) is an extraction probability. It is not identified with black-hole age, radiated entropy, or evaporation time. No passive evaporation, horizon state, gravitational factorization, backreaction estimate, or microscopic gravity dictionary is supplied. \section{Notation and referenced channel metrics} \label{sec:metrics} For a finite matrix \(X\), write \[ \norm X_1=\Tr\sqrt{X^\dagger X}, \qquad \norm X_\infty=\text{operator norm}. \] For states, \[ \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1. \] For channels with the same input and output spaces, \begin{equation} \ddiam(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond. \label{eq:diamond} \end{equation} For a difference of finite-dimensional channels, this equals the supremum of the output trace distance over referenced input states. A reference of input dimension suffices. These are standard finite-channel conventions \cite{Watrous}. Let \(M\) be a logical qubit, \(Q\) an external reference, \(R\) the accessible two-mode output, and \(E\) the residual two-mode output. The decoded qubit is \(M'\). All logarithms in entropy and entanglement expressions are base two. For a qubit-output channel \(\Lambda\), define \[ \ket{\Phi}=\frac{\ket{00}+\ket{11}}{\sqrt2}, \qquad F_e(\Lambda)= \bra\Phi(\id_Q\otimes\Lambda)(\kb\Phi\Phi)\ket\Phi. \] For a purification \(\ket{\psi_\rho}_{QM}\) of \(\rho_M\), define \[ F_{\mathrm{wc}}(\Lambda) = \inf_{\rho_M} \bra{\psi_\rho} (\id_Q\otimes\Lambda)(\kb{\psi_\rho}{\psi_\rho}) \ket{\psi_\rho}. \] The target overlaps are squared fidelities against their pure targets. For an accessible channel \(\NN:M\to R\), set \begin{equation} \ee_{\mathrm{rec}}(\NN) = \inf_{\DD:R\to M'\ {\rm CPTP}} \ddiam(\DD\NN,\id_M), \label{eq:optimal} \end{equation} where the input and decoded logical bases are identified. A replacement channel is \[ \RR_\tau(X)=\Tr(X)\tau, \] and \(\Ad_U(X)=UXU^\dagger\). \begin{lemma}[Elementary contraction and effect bounds] \label{lem:contraction} A CPTP map contracts trace norm on Hermitian inputs. Consequently trace distance contracts under channels, and channel half-diamond distance contracts under fixed CPTP postprocessing. For states \(\rho,\sigma\) and an effect \(0\le F\le\Id\), \[ |\Tr(F(\rho-\sigma))|\le\dtr(\rho,\sigma). \] \end{lemma} \begin{proof} For the Hermitian Jordan decomposition \(X=X_+-X_-\), positivity and trace preservation give \[ \norm{\Phi(X)}_1 \le \Tr\Phi(X_+)+\Tr\Phi(X_-) =\norm X_1. \] The same argument applies with a reference because a CPTP map remains positive after tensoring with its identity. For the effect bound, the trace-zero difference \(\rho-\sigma\) has positive and negative parts of equal trace \(\norm{\rho-\sigma}_1/2\). Testing either part against \(F\) proves the assertion. \end{proof} \section{The complete signed extraction channel} \label{sec:channel} \subsection{Occupation ordering and the declared isometry} Set \[ \Hil_R=\C^4,\qquad \Hil_E=\C^4, \] with ordered occupation bases \[ \ket{00},\ket{01},\ket{10},\ket{11}. \] The two occupations in each subsystem refer to two distinct species or modes. The fixed tensor representation is bath first, residual second. Let \[ 0\le\eta\le1,\qquad t=1-\eta. \] Here \(t\) is a residual probability, not a time variable. \begin{definition}[Phase-corrected signed extractor] \label{def:extractor} The map \(V_\eta:\C^2\to\Hil_R\otimes\Hil_E\) is \begin{align} V_\eta\ket0 &=\ket{00}_R\ket{00}_E, \label{eq:Vzero}\\ V_\eta\ket1 &=\eta\ket{11}_R\ket{00}_E +\sqrt{\eta t}\ket{10}_R\ket{01}_E \nonumber\\ &\quad-\sqrt{\eta t}\ket{01}_R\ket{10}_E +t\ket{00}_R\ket{11}_E. \label{eq:Vone} \end{align} Define the accessible and complete complementary channels by \[ \NN_\eta(X)=\Tr_E(V_\eta XV_\eta^\dagger), \qquad \CC_\eta(X)=\Tr_R(V_\eta XV_\eta^\dagger). \] \end{definition} \begin{remark}[Fermionic signs and logical parity] The minus sign in \eqref{eq:Vone} is the bath-first CAR sign. For residual creators \(e_1^\dagger,e_2^\dagger\) and bath creators \(b_1^\dagger,b_2^\dagger\), expand \[ (\sqrt t\,e_1^\dagger+\sqrt\eta\,b_1^\dagger) (\sqrt t\,e_2^\dagger+\sqrt\eta\,b_2^\dagger) \ket{\mathrm{vac}}. \] Moving bath creators before residual creators gives precisely \eqref{eq:Vone}. Both logical basis vectors have even total fermionic parity. The reference \(Q\) is an abstract logical register, not an additional occupation mode. All partial traces use the specified occupation representation. \end{remark} \begin{remark}[Relation to a complex transfer amplitude] If an uncorrected amplitude is \(q=\sqrt\eta e^{\ii\alpha}\), the occupied column has coefficients \(q^2,q\sqrt t,-q\sqrt t,t\). Applying the accessible phase correction \(e^{-\ii\alpha N_R}\) gives Definition~\ref{def:extractor}. The phase is fixed independently of the logical input. The ideal-phase results below do not certify an estimated or implemented phase correction. \end{remark} \subsection{Kraus operators and the complete complement} \begin{theorem}[Physicality and explicit channels] \label{thm:channels} The map \(V_\eta\) is an isometry for every \(\eta\in[0,1]\). Both channels in Definition~\ref{def:extractor} are CPTP. A Kraus family for \(\NN_\eta\), indexed by residual occupation, is \begin{align} A_{00}&=\kb{00}{0}+\eta\kb{11}{1},& A_{01}&=\sqrt{\eta t}\kb{10}{1},\nonumber\\ A_{10}&=-\sqrt{\eta t}\kb{01}{1},& A_{11}&=t\kb{00}{1}. \label{eq:kraus} \end{align} The complete complement is \begin{equation} \CC_\eta(X) = \sum_{a,b\in\{00,01,10,11\}} \Tr(A_aXA_b^\dagger)\kb a b_E. \label{eq:complement} \end{equation} These formulas retain arbitrary external references. \end{theorem} \begin{proof} The two columns of \(V_\eta\) are orthogonal, and the occupied column has squared norm \[ \eta^2+2\eta t+t^2=(\eta+t)^2=1. \] Thus \(V_\eta^\dagger V_\eta=\Id_2\). Reading its coefficients in the residual basis gives \eqref{eq:kraus}. Direct calculation gives \[ \sum_aA_a^\dagger A_a =\diag(1,\eta^2+2\eta t+t^2)=\Id_2. \] The accessible map is therefore CPTP. The second partial trace of the same isometry is CPTP and gives \eqref{eq:complement}. Tensoring the defining isometry with \(\Id_Q\) proves the reference statement. \end{proof} \begin{remark}[Fixed spectator systems] Tensoring \(V_\eta\) with a fixed pure spectator state does not change any formula below. The spectator remains part of the complete environment, but adds no entropy or logical dependence. An input-correlated or mixed spectator is not covered by this simplification without retaining its preparation purification. \end{remark} \section{Excitation, coherence, and subsystem purity} \label{sec:diagnostics} Write an arbitrary logical density matrix as \begin{equation} \rho_M= \begin{pmatrix} 1-p&z\\z^*&p \end{pmatrix}, \qquad 0\le p\le1,\qquad |z|^2\le p(1-p). \label{eq:input} \end{equation} \begin{theorem}[Exact marginal matrices and diagnostics] \label{thm:marginals} In the declared occupation bases, \begin{align} \rho_R=\NN_\eta(\rho_M) &= \begin{pmatrix} 1-p+pt^2&0&0&\eta z\\ 0&p\eta t&0&0\\ 0&0&p\eta t&0\\ \eta z^*&0&0&p\eta^2 \end{pmatrix}, \label{eq:rhoR}\\ \rho_E=\CC_\eta(\rho_M) &= \begin{pmatrix} 1-p+p\eta^2&0&0&t z\\ 0&p\eta t&0&0\\ 0&0&p\eta t&0\\ t z^*&0&0&pt^2 \end{pmatrix}. \label{eq:rhoE} \end{align} For \[ N_R=N_E=\diag(0,1,1,2) \] on the respective subsystems, \begin{equation} \boxed{ \langle N_R\rangle=2p\eta,\qquad \langle N_E\rangle=2pt. } \label{eq:occupation} \end{equation} The distinguished occupation-basis coherences are \begin{equation} |(\rho_R)_{00,11}|=\eta|z|, \qquad |(\rho_E)_{00,11}|=t|z|. \label{eq:coherence} \end{equation} The accessible purity is \begin{equation} \boxed{ P_R= (1-p+pt^2)^2+p^2\eta^4 +2p^2\eta^2t^2+2\eta^2|z|^2. } \label{eq:purity} \end{equation} The residual purity follows by interchanging \(\eta\) and \(t\). \end{theorem} \begin{proof} Insert \eqref{eq:input} into the Kraus expressions. Only \(A_{00}\) contributes the accessible coherence. For the complement, the only off-diagonal residual matrix units are \(00,11\) and their adjoints, with coefficient \(tz\). This gives \eqref{eq:rhoR} and \eqref{eq:rhoE}. Tracing either marginal against its number operator gives \[ 2p\eta t+2p\eta^2=2p\eta, \qquad 2p\eta t+2pt^2=2pt. \] The coherence formulas are their displayed entries. For a Hermitian matrix, \(\Tr\rho^2=\sum_{i,j}|\rho_{ij}|^2\). Applying this identity to \eqref{eq:rhoR} gives \eqref{eq:purity}. \end{proof} \begin{corollary}[Global retention versus marginal loss] \label{cor:global} For every logical state, \[ \langle N_R+N_E\rangle=2p. \] The nonzero eigenvalues of \(V_\eta\rho_MV_\eta^\dagger\) are those of \(\rho_M\). In particular, \[ S(RE)=S(M). \] For every purified input, the joint referenced output is pure. Thus the declared isometry does not globally destroy information. \end{corollary} \begin{proof} The occupation statement follows from \(\eta+t=1\). An isometry preserves inner products and embeds the spectral decomposition of its input with the same nonzero eigenvalues. Tensoring with a reference preserves the norm and purity of a purified input. \end{proof} \begin{remark}[Different diagnostic content] Occupation means depend only on \(p\). The purity also depends on \(|z|^2\), and the coherence depends on the declared basis. A single input's marginal purity or coherence is not a criterion for recovery of all referenced inputs. Loss of accessible information means failure of recovery from \(R\), not nonunitarity of the retained \(RE\) experiment. \end{remark} \section{Bell-input reference correlations} \label{sec:reference} Set \[ \ket{\Psi_\eta}_{QRE} =(\Id_Q\otimes V_\eta)\ket{\Phi}_{QM}. \] The logical marginal is \(I_M/2\), so \(p=1/2\) and \(z=0\) in the marginal formulas. \begin{theorem}[Spectra and mutual-information balance] \label{thm:information} For the Bell input, \begin{equation} \spec(\rho_R) = \left\{ \frac{1+t^2}{2},\ \frac{\eta^2}{2},\ \frac{\eta t}{2},\ \frac{\eta t}{2} \right\}, \label{eq:BellSpectrum} \end{equation} including zero multiplicities. The residual spectrum is obtained by interchanging \(\eta\) and \(t\). With \[ S(X)=-\Tr(\rho_X\log_2\rho_X), \qquad I(X:Y)=S(X)+S(Y)-S(XY), \] using \(0\log_2 0=0\), one has \begin{align} I(Q:R)&=1+S(R)-S(E),\label{eq:MIr}\\ I(Q:E)&=1+S(E)-S(R),\label{eq:MIe}\\ \boxed{I(Q:R)+I(Q:E)=2.}\label{eq:balance} \end{align} Consequently, \[ \begin{array}{c|cc} \eta&I(Q:R)&I(Q:E)\\ \hline 0&0&2\\ 1/2&1&1\\ 1&2&0 \end{array} \] and \[ P_R(\eta=0)=1,\qquad P_R(\eta=1/2)=7/16,\qquad P_R(\eta=1)=1/2. \] \end{theorem} \begin{proof} For the Bell input the matrices \eqref{eq:rhoR} and \eqref{eq:rhoE} are diagonal, giving the spectra. The global state \(\ket{\Psi_\eta}\) is pure and \(S(Q)=1\). For a pure bipartite state the two reduced states have the same nonzero spectrum: expand its coefficient matrix and compare the nonzero eigenvalues of \(AA^\dagger\) and \(A^\dagger A\). Apply this to \(QR|E\) and \(QE|R\) to obtain \(S(QR)=S(E)\) and \(S(QE)=S(R)\). This proves \eqref{eq:MIr}--\eqref{eq:balance}. At the endpoints the spectra give entropies \(0,1\) in opposite orders. At \(\eta=t=1/2\), the spectra coincide. Squaring and summing the accessible eigenvalues gives the three displayed purities. \end{proof} \begin{remark}[No duplication theorem is claimed] Equation \eqref{eq:balance} is a pure-state mutual-information identity. Mutual information includes classical correlations. It is neither a statement that both subsystems contain an independently recoverable qubit nor a gravitational entropy law. \end{remark} \subsection{Entanglement before decoding} For an ordinary bipartite density matrix, define negativity by \[ \mathcal N^{Q|R}(\rho) =\frac{\norm{\rho^{T_Q}}_1-1}{2}, \] with partial transpose in the declared reference basis \cite{VidalWerner}. \begin{proposition}[Exact reference--accessible negativity] \label{prop:rawneg} For the Bell-input state, \begin{equation} \boxed{ \mathcal N^{Q|R}(\rho_{QR}) = \frac{\sqrt{t^4+4\eta^2}-t^2}{4}. } \label{eq:rawneg} \end{equation} Thus the reference and accessible output are entangled for every \(\eta>0\). \end{proposition} \begin{proof} The only coherence of \(\rho_{QR}\) connects \(\ket0_Q\ket{00}_R\) and \(\ket1_Q\ket{11}_R\), with coefficient \(\eta/2\). Under \(T_Q\), it enters the block on \(\ket0_Q\ket{11}_R,\ket1_Q\ket{00}_R\): \[ \begin{pmatrix} 0&\eta/2\\ \eta/2&t^2/2 \end{pmatrix}. \] Its eigenvalues are \[ \frac{t^2\pm\sqrt{t^4+4\eta^2}}4. \] All remaining eigenvalues are nonnegative diagonal entries or zeros. The negative eigenvalue therefore gives \eqref{eq:rawneg}. A separable state's partial transpose is positive, because transposition preserves positivity of each factor in a separable decomposition. The strictly negative eigenvalue at \(\eta>0\) proves entanglement. \end{proof} This proposition concerns \(Q|R\), not the partition between the two bath modes. Reference entanglement can survive even when exact recovery is impossible. \section{A complete specified decoder and optimal-recovery boundaries} \label{sec:decoder} Let \[ W:\C^2\to\Hil_R,\qquad W\ket0=\ket{00},\quad W\ket1=\ket{11}, \qquad P=WW^\dagger. \] \begin{definition}[Specified complete decoder] Define \begin{equation} \DD_\star(Y) =W^\dagger YW+\Tr((\Id_R-P)Y)\frac{\Id_2}{2}. \label{eq:decoder} \end{equation} Every off-code event is retained and mapped to the maximally mixed decoded qubit. \end{definition} \begin{lemma}[Decoder physicality] The map \(\DD_\star\) is CPTP on every input state on \(\Hil_R\). \end{lemma} \begin{proof} Use the retained Kraus operator \(W^\dagger\), together with \[ \frac1{\sqrt2}\kb i a, \qquad i=0,1,\quad a=01,10. \] Their effects sum to \(P+\kb{01}{01}+\kb{10}{10}=\Id_R\). Their Kraus sum is exactly \eqref{eq:decoder}. \end{proof} \begin{theorem}[Exact specified-decoder performance] \label{thm:specified} The decoded channel is \begin{equation} \boxed{ \Lambda_\eta:=\DD_\star\NN_\eta =\eta\,\id+(1-\eta)\RR_{\kb00}. } \label{eq:decoded} \end{equation} It satisfies \begin{align} \ddiam(\Lambda_\eta,\id)&=1-\eta,\label{eq:specifiedError}\\ F_e(\Lambda_\eta)&=\frac{1+3\eta}{4},\label{eq:BellFid}\\ F_{\mathrm{wc}}(\Lambda_\eta)&=\eta.\label{eq:WorstFid} \end{align} For the Bell-input decoded state, \begin{equation} \omega_{QM'} = \eta\kb\Phi\Phi+t\frac{\Id_Q}{2}\otimes\kb00, \label{eq:decodedBell} \end{equation} the two-qubit concurrence is \begin{equation} \boxed{C(Q:M')=\eta.} \label{eq:decodedConc} \end{equation} No optimality of \(\DD_\star\) is asserted for intermediate \(\eta\). \end{theorem} \begin{proof} The odd occupations in \eqref{eq:rhoR} contribute \(p\eta t\Id_2\) after decoding. The decoded lower-right entry is \(p\eta^2+p\eta t=p\eta\), and its off-diagonal entry is \(\eta z\). The upper-left entry is \(1-p\eta\). This proves \eqref{eq:decoded}. A half-diamond distance between channels is at most one. The identity and replacement by \(\ket0\) attain distance one on input \(\ket1\). Homogeneity therefore gives \eqref{eq:specifiedError}. The replacement term's Bell overlap is \(1/4\), proving \eqref{eq:BellFid}. For any purified input, the target overlap of \eqref{eq:decoded} is at least \(\eta\), because the replacement contribution is nonnegative. Input \(\ket1\) gives zero replacement contribution and attains \(\eta\), proving \eqref{eq:WorstFid}. For concurrence, use Wootters' definition \cite{Wootters}: the concurrence is the positive part of the largest square-root eigenvalue of \(\omega\widetilde\omega\) minus the other three, where \[ \widetilde\omega=(Y\otimes Y)\overline\omega(Y\otimes Y). \] In the ordered basis \(00,01,10,11\), \eqref{eq:decodedBell} is \[ \omega= \begin{pmatrix} 1/2&0&0&\eta/2\\ 0&0&0&0\\ 0&0&t/2&0\\ \eta/2&0&0&\eta/2 \end{pmatrix}. \] The square-root eigenvalues are \[ \frac{\sqrt\eta+\eta}{2},\qquad \frac{\sqrt\eta-\eta}{2},\qquad0,\qquad0. \] Their difference is \(\eta\). \end{proof} \subsection{What can be said about all decoders} \begin{proposition}[No exact intermediate recovery and endpoint optima] \label{prop:optimal} For \(0\le\eta<1\), \begin{equation} \ee_{\mathrm{rec}}(\NN_\eta)\ge\frac{(1-\eta)^2}{2}>0. \label{eq:elementaryLower} \end{equation} For all \(\eta\), \[ \ee_{\mathrm{rec}}(\NN_\eta) \le\min\{1-\eta,3/4\}. \] At the endpoints, \begin{equation} \boxed{ \ee_{\mathrm{rec}}(\NN_0)=3/4,\qquad \ee_{\mathrm{rec}}(\NN_1)=0. } \label{eq:endpointOptimal} \end{equation} In particular, the specified decoder's error one at \(\eta=0\) is not the optimal error. \end{proposition} \begin{proof} Let \(\rho_i=\NN_\eta(\kb ii)\). These states are diagonal and \[ \dtr(\rho_0,\rho_1)=1-t^2. \] For any decoder with error \(\ee\), its decoded basis inputs are within \(\ee\) of their respective orthogonal targets. The triangle inequality and contraction imply \[ 1\le 2\ee+\dtr(\rho_0,\rho_1)=2\ee+1-t^2. \] Thus \(\ee\ge t^2/2\). The specified decoder gives the upper bound \(1-\eta\). Another decoder ignores its input and returns \(\Id_2/2\). For Pauli matrices \(X,Y,Z\), \[ \RR_{\Id_2/2} =\frac14(\id+\Ad_X+\Ad_Y+\Ad_Z). \] The triangle inequality gives its distance to the identity at most \(3/4\). At \(\eta=0\), every decoded channel is a replacement \(\RR_\tau\). Its Bell output is \(\Id_Q/2\otimes\tau\), whose overlap with \(\ket\Phi\) is \(1/4\), independently of \(\tau\). The Bell-projector effect and Lemma~\ref{lem:contraction} give distance at least \(3/4\). The maximally mixed replacement attains the preceding upper bound. At \(\eta=1\), \(\NN_1=\Ad_W\), and the specified decoder is its left inverse. \end{proof} \begin{remark}[Purity endpoints] At \(\eta=0\), the accessible subsystem is pure vacuum for every message, but the optimal recovery error is \(3/4\). At \(\eta=1\), a Bell input produces an accessible marginal \(W(\Id_2/2)W^\dagger\) of purity \(1/2\), but recovery is exact. At full transfer the residual subsystem is pure vacuum. That purity certifies neither residual recovery nor global loss: the message has moved to \(R\). \end{remark} \section{Population-preserving dephasing with its complete environment} \label{sec:dephasing} The original extractor is now followed by a separately specified noise channel. It is a different experiment, not a hidden property of Definition~\ref{def:extractor}. \subsection{Physical output operation and dilation} On the entire accessible space define \[ S_R=\diag(1,1,1,-1). \] It is Hermitian, unitary, parity preserving, and commutes with every occupation projector and with \(N_R\). For \(0\le\lambda\le1\), put \[ a_\lambda=\frac{1+\lambda}{2},\qquad b_\lambda=\frac{1-\lambda}{2}, \] and define \begin{equation} \mathcal Z^R_\lambda(Y) =a_\lambda Y+b_\lambda S_RYS_R. \label{eq:dephaser} \end{equation} Let \(F=\C^2\) be an additional environmental register. The isometry \begin{equation} U_\lambda x = \sqrt{a_\lambda}\,x\otimes\ket0_F +\sqrt{b_\lambda}\,S_Rx\otimes\ket1_F \label{eq:dephasingDilation} \end{equation} is a dilation of \(\mathcal Z^R_\lambda\). Define \[ \widetilde V_{\eta,\lambda} =(U_\lambda\otimes\Id_E)V_\eta, \] with the fixed output reordering \(RFE\to REF\). The new accessible channel is \[ \NN_{\eta,\lambda}=\mathcal Z^R_\lambda\NN_\eta. \] Its complete environment is \(EF\), not merely \(E\). \begin{proposition}[Complete noisy channel and unchanged populations] \label{prop:noisy} A Kraus family for \(\NN_{\eta,\lambda}\) is \[ L_{a,0}=\sqrt{a_\lambda}A_a,\qquad L_{a,1}=\sqrt{b_\lambda}S_RA_a. \] Its complete complement is \[ \CC_{\eta,\lambda}(X) = \sum_{a,s,b,u} \Tr(L_{a,s}XL_{b,u}^\dagger) \kb{a,s}{b,u}_{EF}. \] The marginal \(\NN_{\eta,\lambda}(\rho_M)\) is \eqref{eq:rhoR} with \(\eta z\) replaced by \(\lambda\eta z\), and its adjoint entry replaced accordingly. All occupation probabilities and \(\langle N_R\rangle\) are unchanged for every input. \end{proposition} \begin{proof} Since \(a_\lambda+b_\lambda=1\) and \(S_R^2=\Id\), the Kraus effects sum to \(\Id_M\). Taking both partial traces of \(\widetilde V_{\eta,\lambda}\) gives the channels and complement. Conjugation by \(S_R\) leaves diagonal entries unchanged and negates the \(00,11\) coherence. The weighted sum multiplies that coherence by \(a_\lambda-b_\lambda=\lambda\). \end{proof} At \(\lambda=0\), the noisy accessible channel has no dependence on \(z\): \[ \NN_{\eta,0}(X) = \langle0|X|0\rangle\,\rho_R^{(0)} +\langle1|X|1\rangle\,\rho_R^{(1)}. \] It is a measure-and-prepare channel. Its referenced output is separable, because it is a sum of positive reference operators tensored with prepared output states. Complete dephasing therefore destroys accessible reference entanglement for every \(\eta\). \subsection{Full-transfer noise: exact optimal recovery} At \(\eta=1\), set \[ u_\lambda =\sqrt{a_\lambda}\ket0+\sqrt{b_\lambda}\ket1, \qquad v_\lambda =\sqrt{a_\lambda}\ket0-\sqrt{b_\lambda}\ket1. \] Their overlap is \(\langle u_\lambda|v_\lambda\rangle=\lambda\). The full-transfer noisy isometry is \begin{align} \widetilde V_{1,\lambda}\ket0 &=W\ket0\otimes\ket{00}_E\otimes u_\lambda,\nonumber\\ \widetilde V_{1,\lambda}\ket1 &=W\ket1\otimes\ket{00}_E\otimes v_\lambda. \label{eq:fullNoisy} \end{align} \begin{theorem}[Exact noisy full-transfer comparison] \label{thm:fullNoise} For the full-transfer dephasing experiment, \begin{equation} \boxed{ \ee_{\mathrm{rec}}(\NN_{1,\lambda}) =\frac{1-\lambda}{2}. } \label{eq:noiseOptimal} \end{equation} The specified decoder attains this optimum. For its Bell-input decoded output, \[ C(Q:M')=\lambda,\qquad F_e=\frac{1+\lambda}{2}. \] The Bell-input accessible marginal is independent of \(\lambda\): \begin{equation} \rho_R=W\frac{\Id_2}{2}W^\dagger, \qquad \langle N_R\rangle=1,\qquad P_R=1/2. \label{eq:identicalMarginal} \end{equation} Writing \[ h_2(x)=-x\log_2x-(1-x)\log_2(1-x), \] with its continuous endpoint values, the mutual informations are \begin{align} I(Q:R)&=2-h_2((1+\lambda)/2),\label{eq:noiseMIr}\\ I(Q:EF)&=h_2((1+\lambda)/2).\label{eq:noiseMIe} \end{align} At complete dephasing, \(\lambda=0\), occupation delivery is perfect, decoded concurrence is zero, and the optimal recovery error is \(1/2\). \end{theorem} \begin{proof} On the logical qubit, the specified decoded channel is \[ \Delta_\lambda =a_\lambda\id+b_\lambda\Ad_Z, \qquad Z=\diag(1,-1). \] The channels \(\Ad_Z\) and \(\id\) have half-diamond distance one: input \(\ket+\) gives orthogonal outputs. Hence \[ \ddiam(\Delta_\lambda,\id)=b_\lambda. \] For the lower bound valid for every decoder, use logical inputs \(\ket+\) and \(\ket-\). Their noisy accessible outputs have trace distance \(\lambda\), whereas their ideal targets have distance one. A decoder of error \(\ee\) must therefore satisfy \[ 1\le2\ee+\lambda. \] Thus \(\ee\ge(1-\lambda)/2\), proving optimality. The decoded Bell state is \[ \omega_\lambda = \frac12\left( \kb{00}{00}+\kb{11}{11} +\lambda\kb{00}{11}+\lambda\kb{11}{00}\right). \] Its overlap with \(\ket\Phi\) is \((1+\lambda)/2\). Its spin-flip square-root eigenvalues are \((1+\lambda)/2,(1-\lambda)/2,0,0\), so its concurrence is \(\lambda\). Tracing the reference in \eqref{eq:fullNoisy} gives \eqref{eq:identicalMarginal}. The \(F\) marginal is \[ \frac12(\kb{u_\lambda}{u_\lambda} +\kb{v_\lambda}{v_\lambda}) =\diag(a_\lambda,b_\lambda). \] Thus \(S(EF)=h_2(a_\lambda)\), while \(S(Q)=S(R)=1\). The joint state \(QREF\) is pure, so the same pure-state entropy argument as in Theorem~\ref{thm:information} gives \eqref{eq:noiseMIr} and \eqref{eq:noiseMIe}. \end{proof} \begin{corollary}[Identical marginal diagnostics do not certify recovery] \label{cor:separation} For the Bell input, the noiseless full-transfer experiment \((\eta,\lambda)=(1,1)\) and the completely dephased experiment \((1,0)\) have identical accessible density matrices. They therefore have identical accessible occupation probabilities, purity, and every other observable depending only on that marginal. Nevertheless their optimal recovery errors are respectively \(0\) and \(1/2\), their decoded reference concurrences are \(1\) and \(0\), and their reference--accessible mutual informations are \(2\) and \(1\). \end{corollary} \begin{proof} Apply Theorem~\ref{thm:fullNoise} at its two endpoints. \end{proof} \begin{remark}[The information is not deleted from the full dilation] At \(\lambda=0\), \(u_0\) and \(v_0\) are orthogonal. For the Bell input, the complete output has the form \[ \frac1{\sqrt2}\left( \ket0_QW\ket0_Ru_0{}_F+ \ket1_QW\ket1_Rv_0{}_F\right)\otimes\ket{00}_E. \] The accessible reference state is only classically correlated. The retained \(RF\) isometry still preserves the input qubit. Discarding \(F\) creates the accessible dephasing channel. Keeping only the original vacuum residual \(E\) would incorrectly hide the environmental information. \end{remark} \section{Certified approximation interfaces} \label{sec:certification} The preceding formulas are exact identities for declared channels. They do not assert a validated amplitude for a particular numerical or continuum experiment. \begin{proposition}[Transmission intervals] \label{prop:interval} If a validated calculation establishes \[ 0\le\eta_-\le\eta\le\eta_+\le1, \] then the ideal-phase specified decoder satisfies \[ e_{\mathrm{specified}}\in[1-\eta_+,1-\eta_-], \] \[ C(Q:M')\in[\eta_-,\eta_+], \qquad F_e\in\left[ \frac{1+3\eta_-}{4},\frac{1+3\eta_+}{4}\right]. \] For a fixed known \(p\), \[ \langle N_R\rangle\in[2p\eta_-,2p\eta_+]. \] Purity can be enclosed by outward evaluation of \eqref{eq:purity}, but it must not be presumed monotone in \(\eta\). \end{proposition} \begin{proof} The first four quantities are affine in \(\eta\) with the displayed monotonicity. The purity is an explicit polynomial in \(\eta,t,p,|z|^2\), so inclusion-preserving interval evaluation encloses its value. \end{proof} \begin{proposition}[Channel-model error and recovery stability] \label{prop:stability} Suppose two accessible channels with the same types obey \[ \ddiam(\widehat\NN,\NN_\eta)\le\xi. \] For any fixed CPTP decoder \(\DD\), \[ \left| \ddiam(\DD\widehat\NN,\id) -\ddiam(\DD\NN_\eta,\id) \right|\le\xi. \] Their optimal recovery errors also satisfy \[ |\ee_{\mathrm{rec}}(\widehat\NN) -\ee_{\mathrm{rec}}(\NN_\eta)|\le\xi. \] For every common referenced input, their output states have trace distance at most \(\xi\). For the corresponding accessible marginals, \[ |\widehat P_R-P_R|\le4\xi, \qquad |\langle\widehat N_R\rangle-\langle N_R\rangle| \le2\xi, \] and \[ |(\widehat\rho_R-\rho_R)_{00,11}|\le\xi. \] For a fixed decoder, the Bell fidelities differ by at most \(\xi\). \end{proposition} \begin{proof} Postprocessing contraction and the reverse triangle inequality prove the fixed-decoder statement. Apply it for every decoder and take infima in both directions to obtain optimal-error stability. The referenced state bound is the definition of channel distance. For purity, \[ |\Tr(\widehat\rho_R^2-\rho_R^2)| \le\norm{\widehat\rho_R-\rho_R}_1 \norm{\widehat\rho_R+\rho_R}_\infty \le4\xi. \] Since the state difference has trace zero, center the occupation operator: \[ |\Tr(N_R(\widehat\rho_R-\rho_R))| =|\Tr((N_R-\Id)(\widehat\rho_R-\rho_R))| \le2\xi, \] because \(\norm{N_R-\Id}_\infty=1\). For distinct basis vectors \(u,v\) and Hermitian \(X\), \[ 2|\bra uX\ket v|\le\norm X_1. \] This follows by testing against a phase-aligned Hermitian off-diagonal operator of norm one. It gives the coherence-entry estimate. The Bell-projector effect bound gives the fidelity estimate. \end{proof} \begin{remark}[What must be certified separately] A transmission interval does not automatically certify: \begin{itemize} \item a physical gravitational identification; \item the implementation of the decoder; \item the accuracy of a phase correction; \item a common identification of residual wavefunctions across different experiments; \item a comparison of their physical complementary channels. \end{itemize} State residual bounds of the kind developed in Em04-A are useful interfaces, but a single-state bound is not automatically a referenced channel bound. A channel certificate must control the specified channel on arbitrary referenced inputs. For an implemented phase error \(\delta\), the noiseless decoded channel instead has the form \[ \eta\Ad_{\diag(1,e^{2\ii\delta})} +t\RR_{\kb00}. \] Its half-diamond error is at most \[ \min\{1,t+\eta|\sin\delta|\}. \] Indeed the relevant unitary channel has distance \(|\sin\delta|\) from the identity: its numerical range has minimal modulus \(|\cos\delta|\), and the pure referenced output distance is the square root of one minus the squared overlap. This phase allowance and any additional implementation error must be charged rather than absorbed into an unverified ideal-phase statement. \end{remark} \section{Interpretation and mathematical conclusion} \label{sec:conclusion} \begin{center} \small \renewcommand{\arraystretch}{1.15} \begin{tabularx}{\linewidth}{@{}p{.29\linewidth}X@{}} \toprule Diagnostic & Question answered\\ \midrule Transferred occupation & How much excitation reaches the accessible modes?\\ Occupation-basis coherence & Which off-diagonal entry survives in the declared basis?\\ Accessible or residual purity & How mixed is that marginal for the chosen input?\\ Reference--accessible negativity & Is the chosen referenced output entangled across \(Q|R\)?\\ Decoded concurrence & How much two-qubit reference entanglement remains after the specified decoder?\\ Mutual information & How much total correlation is present for the chosen input?\\ Specified half-diamond error & How well does this complete decoder work on all referenced inputs?\\ Optimal recovery error & How well can any deterministic CPTP decoder work?\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[Diagnostic separation] \label{thm:release} For the declared extraction and output-noise experiments: \begin{enumerate}[label=\textup{(\roman*)}] \item The signed extraction map and its complete complement are physical for every \(\eta\in[0,1]\). \item Occupation, coherence, purity, Bell-input spectra, reference mutual information, and reference negativity obey the proved identities. \item The specified decoder has error \(1-\eta\), Bell fidelity \((1+3\eta)/4\), worst-case target overlap \(\eta\), and decoded concurrence \(\eta\). \item Exact recovery is impossible at \(\eta<1\), but the specified decoder is not presumed optimal there. Optimal recovery errors at \(\eta=0,1\) are \(3/4,0\). \item Population-preserving dephasing retains an additional complete environment. At full transfer its optimal recovery error is \((1-\lambda)/2\), although the Bell-input accessible marginal is independent of \(\lambda\). \item Accessible marginal purity, occupation delivery, and even knowledge of the complete accessible marginal for a single Bell-input experiment do not replace a referenced channel-recovery criterion. \end{enumerate} These are finite-channel information-flow results. They do not establish black-hole evaporation, a gravitational Page curve, or a solution to the black-hole information paradox. \end{theorem} \begin{proof} Theorem~\ref{thm:channels} proves (i). Theorem~\ref{thm:marginals}, Theorem~\ref{thm:information}, and Proposition~\ref{prop:rawneg} prove (ii). Theorem~\ref{thm:specified} proves (iii). Proposition~\ref{prop:optimal} proves (iv). Proposition~\ref{prop:noisy} and Theorem~\ref{thm:fullNoise} prove (v). The purity endpoints and Corollary~\ref{cor:separation} prove (vi). None of the hypotheses specifies a gravitational identification. \end{proof} The natural application is an exactly solvable benchmark for information-flow diagnostics and recovery certification. A sequential evaporation-like model would require newly specified fresh output registers, an access schedule, internal dynamics, and a complete environment at every step. Such a construction is not obtained by relabeling \(\eta\) as time. \appendix \section{Continuum interpretation and endpoint attainability} \label{app:continuum} The finite family includes \(\eta=1\) as an exact mathematical endpoint. It does not imply that the retained-storage continuum experiment of Em03-H attains that endpoint. For clarity, the obstruction can be restated independently. Let \[ \mathfrak h=L^2(\R_+),\qquad (h\psi)(\omega)=\omega\psi(\omega),\qquad f(\omega)=\sqrt2e^{-\omega}. \] Take \(t_0\ge0\), a initially vacant bath level, and the initial one-species state \[ \psi(0,\omega)=e^{-\ii\omega t_0}f(\omega), \qquad B(0)=0. \] For finite extraction time \(T\), use bounded piecewise continuous real controls \(v(u),\kappa(u)\) and the generator \[ K(u)= \begin{pmatrix} h+v(u)\kb ff&\kappa(u)\ket f\\ \kappa(u)\bra f&1 \end{pmatrix} \] on \(\mathfrak h\oplus\C\). Relative to \(h\oplus1\), the perturbation is bounded, self-adjoint, and finite rank. Its interaction-picture Dyson series gives a unitary propagator. Put \[ A(u)=\langle f,\psi(u)\rangle,\qquad c(u)=v(u)A(u)+\kappa(u)B(u). \] Variation of constants gives \[ \psi(T,\omega) = f(\omega)e^{-\ii\omega(T+t_0)} -\ii f(\omega)\int_0^T e^{-\ii\omega(T-s)}c(s)\,\mathrm ds. \] \begin{proposition}[No exact finite-time continuum transfer] \label{prop:noexact} In this declared continuum model, \(|B(T)|<1\) for finite \(T\). \end{proposition} \begin{proof} If \(|B(T)|=1\), unitarity forces \(\psi(T)=0\). Since \(f(\omega)>0\), the packet formula implies \[ e^{-\ii\omega t_0} = \ii\int_0^T e^{\ii\omega s}c(s)\,\mathrm ds \] for almost every positive \(\omega\). The function \(c\) is bounded and integrable. Both sides extend to entire functions. Their equality on a full-measure subset of a real interval gives equality everywhere by continuity and the identity theorem. At \(\omega=\ii y\), \(y>0\), the left side is \(e^{yt_0}\), whereas \[ \left|\int_0^T e^{-ys}c(s)\,\mathrm ds\right| \le\frac{\norm c_{L^\infty}}{y}. \] The right side tends to zero and the left side is at least one, a contradiction. \end{proof} This proposition supplies neither a uniform positive numerical error floor nor a quarter-error exclusion. It distinguishes physical endpoint attainability in this model from endpoint identities of the abstract channel family. No numerical value of \(B(T)\) is asserted here. \section{Analytical corrections and submission checks} \label{app:checks} The construction incorporates the following corrections to the motivating proposal: \begin{enumerate} \item The transfer family is stated independently of its continuum realization. Full transfer is not falsely reported as a finite-time continuum result. \item The specified decoder's zero-transfer error is one, whereas the optimal deterministic error is \(3/4\). \item The Bell-input spectra and mutual-information values are labeled as input-specific. \item The complete noisy complement includes the dephasing register \(F\). \item At full transfer, complete dephasing has optimal recovery error \(1/2\), proved for all deterministic decoders. \item Reference concurrence is calculated using the spin-flip spectrum, not identified with a bath-mode entanglement measure. \item The extraction parameter is not assumed monotone along a physical time trajectory. \item The Knill--Laflamme journal citation uses the publisher's \emph{Physical Review A} article, rather than the conflicting journal metadata displayed on the supplied arXiv landing page. \end{enumerate} Before claiming an executed verification package, independently: \begin{itemize} \item compile until all references stabilize and inspect the PDF; \item construct the isometry matrices directly and check \(V_\eta^\dagger V_\eta=\Id\); \item trace the full referenced states independently of the displayed marginal formulas; \item compare spin-flip and partial-transpose spectra with the analytical expressions, including \(\eta=0,1\); \item verify the noisy isometry with all \(EF\) registers retained; \item retain input data, package versions, tolerances, raw outputs, and failures for every executed numerical test. \end{itemize} These are proposed checks, not execution claims. Floating-point agreement is not a rigorous certificate. \section*{Declarations} \paragraph{Availability.} This single source contains the model, proofs, and internal bibliography. No external data are required for typesetting. \paragraph{Execution.} No compilation, numerical calculation, independent symbolic execution, or formal proof checking is claimed. \paragraph{Physical interpretation.} The work is a finite-channel benchmark motivated by information-flow questions. It does not identify the residual subsystem with a black-hole interior, the accessible bath with Hawking radiation, or either parameter with gravitational evolution. \paragraph{Attribution.} Quantum recovery, complementary channels, entropy identities, concurrence, negativity, and dephasing are established frameworks. The model-specific calculations are proved here without claiming a new general foundation or exhaustive novelty verification. \begin{thebibliography}{99} \bibitem{Avery} S.~G.~Avery, ``Qubit Models of Black Hole Evaporation,'' arXiv:\texttt{1109.2911v2} [hep-th] (2012). Used as context for finite-dimensional evaporation models, not as a derivation of the present extractor. \bibitem{HaydenPreskill} P.~Hayden and J.~Preskill, ``Black holes as mirrors: quantum information in random subsystems,'' \emph{Journal of High Energy Physics} \textbf{09} (2007), 120. DOI: \texttt{10.1088/1126-6708/2007/09/120}. arXiv:\texttt{0708.4025}. Used as recovery-task precedent under its own dynamical and access assumptions. \bibitem{KnillLaflamme} E.~Knill and R.~Laflamme, ``Theory of quantum error-correcting codes,'' \emph{Physical Review A} \textbf{55}, 900--911 (1997). DOI: \texttt{10.1103/PhysRevA.55.900}. Preprint title: ``A Theory of Quantum Error-Correcting Codes,'' arXiv:\texttt{quant-ph/9604034}. Used for the established reference-based exact-recovery framework. \bibitem{KSW} D.~Kretschmann, D.~Schlingemann, and R.~F.~Werner, ``The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation,'' \emph{IEEE Transactions on Information Theory} \textbf{54}, 1708--1717 (2008). DOI: \texttt{10.1109/TIT.2008.917696}. arXiv:\texttt{quant-ph/0605009}. Used as information--disturbance and channel-continuity background. \bibitem{BenyOreshkov} C.~B\'eny and O.~Oreshkov, ``General conditions for approximate quantum error correction and near-optimal recovery channels,'' \emph{Physical Review Letters} \textbf{104}, 120501 (2010). DOI: \texttt{10.1103/PhysRevLett.104.120501}. arXiv:\texttt{0907.5391}. Used as approximate-recovery background; its fidelity metric is not silently identified with the half-diamond metric used here. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}, Cambridge University Press (2018). Used for finite channels, referenced channel norms, and entropy conventions. \bibitem{Wootters} W.~K.~Wootters, ``Entanglement of formation of an arbitrary state of two qubits,'' \emph{Physical Review Letters} \textbf{80}, 2245--2248 (1998). DOI: \texttt{10.1103/PhysRevLett.80.2245}. arXiv:\texttt{quant-ph/9709029}. Used for the concurrence definition. \bibitem{VidalWerner} G.~Vidal and R.~F.~Werner, ``Computable measure of entanglement,'' \emph{Physical Review A} \textbf{65}, 032314 (2002). DOI: \texttt{10.1103/PhysRevA.65.032314}. arXiv:\texttt{quant-ph/0102117}. Used for the negativity definition. \end{thebibliography} \end{document}