\documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{microtype} \usepackage{booktabs,tabularx,longtable,array} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={Hawk04-B: Quantum Recovery, Destructive Readout, and Complete Residuals}, pdfauthor={Ngo Cheung}, pdfsubject={Reference-preserving quantum channels and typed finite programs}, pdfkeywords={quantum recovery, complementary channel, no-hiding, destructive measurement, typed quantum programs, regulated reservoir} } \setlength{\emergencystretch}{3em} \setlength{\tabcolsep}{5pt} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \newcolumntype{L}[1]{>{\raggedright\arraybackslash}p{#1}} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \numberwithin{equation}{section} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\ii}{\mathrm i} \newcommand{\dd}{\mathrm d} \newcommand{\id}{\mathrm{id}} \newcommand{\NN}{\mathcal N} \newcommand{\CC}{\mathcal C} \newcommand{\DD}{\mathcal D} \newcommand{\TT}{\mathcal T} \newcommand{\VV}{\mathcal V} \newcommand{\MM}{\mathcal M} \newcommand{\FS}{\mathcal F_{\mathrm s}} \newcommand{\FF}{\mathcal F_{\mathrm f}} \newcommand{\erec}{\varepsilon_{\mathrm{rec}}} \newcommand{\dleak}{\delta_{\mathrm{full}}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddia}{d_{\diamond}} \newcommand{\Qtype}{\mathsf Q} \newcommand{\Bit}{\mathsf{Bit}} \newcommand{\Itype}{\mathsf I} \newcommand{\New}{\mathsf{new}} \newcommand{\Enter}{\mathsf{Enter}} \newcommand{\Drop}{\mathsf{dropBit}} \newcommand{\ReturnQ}{\mathsf{returnQ}} \newcommand{\ReturnBit}{\mathsf{returnBit}} \newcommand{\ForgetQ}{\mathsf{forgetQ}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\ket{#1}\bra{#2}} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\diag}{diag} \title{ Hawk04-B\\[0.3em] Black-Hole Information as an Access Problem: Enter, Quantum Recovery, and Complete Residuals } \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We distinguish complete unitary preservation, subsystem quantum recovery, destructive observation, and operational discard in a reference-preserving channel formulation. A finite acyclic typed fragment represents allocation, specified unitary transformations, destructive computational readout, and classical forgetting. Quantum recovery returns a quantum register; terminal readout returns a classical record and is a different task. For a complete isometry \(V:C\to R\otimes B\), we give an explicit global recovery channel and prove that exact recovery from \(R\) is equivalent to constancy of the complete complementary channel to \(B\). A process retaining only classical outcomes cannot reconstruct an arbitrary \(d\)-dimensional quantum code with half-diamond error below \(1-d^{-1}\). Unobserved readout of \(B\), followed by discarding its outcomes, is exactly partial trace over \(B\) and leaves the radiation channel unchanged. Its coherent dilation retains the information omitted from the classical interface. Independently, a finite-band quartic source--scalar-reservoir Hamiltonian is constructed and approximated by finite-dimensional proxies with explicit complete-channel allowances. Separate qubit dictionaries preserve the accessible--inaccessible partition. For a commuting four-Majorana control, the exact errors and complete residual leakage are \[ \erec^R(t)=\frac{3-\sqrt{1-c_R(t)^2}}4,\qquad \erec^B(t)=\frac{1-c_R(t)}2,\qquad \dleak(t)=\max\left\{\frac12,\frac{1+2c_R(t)}4\right\}. \] Thus the complete output is reversible although neither output alone need recover the diary. Access-preserving endpoint continuity and fixed-proxy recurrence are proved separately from syntactic termination. The gravitational implication remains conditional on a same-model unitary dilation, outgoing access construction, and constant endpoint residual. No positive noncommuting recovery window, physical field-to-memory receiver, quantum-gravity unitarity theorem, or gravitational evaporation endpoint is established. \end{abstract} \begin{center} \fbox{\begin{minipage}{0.93\linewidth} \textbf{The governing distinctions.} \[ \begin{gathered} \text{complete preservation} \ \not\Longrightarrow\ \text{recovery from a chosen subsystem},\\ \text{quantum recovery} \ \ne\ \text{destructive classical observation},\\ \text{operational discard} \ \ne\ \text{proved physical annihilation},\\ \text{finite-program termination} \ \ne\ \text{a physical evaporation endpoint}. \end{gathered} The residual is an output ledger, not a deletion mechanism. \end{minipage}} \end{center} \tableofcontents %===================================================================== \section{Introduction} \label{sec:introduction} The black-hole information problem asks whether a proposed evaporation description is compatible with complete quantum evolution. Hawking's radiation calculation and information-loss proposal provide its historical setting \cite{Hawking1975,Hawking1976}. A finite circuit semantics does not derive that physical evolution. This paper supplies an operational formulation in which the following questions have distinct mathematical objects: \begin{enumerate}[label=\textup{(\roman*)}] \item Is the finite quantum code preserved by the complete evolution? \item Does the declared accessible subsystem admit a quantum decoder? \item What instrument produces a specified classical record? \item Which outputs are omitted from the observer's description? \item Does the same physical model have an access-preserving endpoint? \end{enumerate} Every code comparison retains arbitrary external references. The exact correction and no-hiding principles are established results \cite{KnillLaflamme,BraunsteinPati}. The approximate channel conversion used here is due to Kretschmann, Schlingemann, and Werner \cite{KSW}. We prove the exact statements locally, identify the imported approximate theorem explicitly, and make no priority claim for these principles. Superoperator semantics for quantum languages also has established precedent \cite{Selinger}. The contribution is the joint accounting of typed destructive readout, complete complementary outputs, access-preserving regulator certificates, and an exact dynamical negative control. In particular, a nonconstant residual is not promoted to a recoverable residual-only copy of the entire unknown code. \subsection{Relation to the supplied Cheung manuscripts} \label{subsec:related} Hawk04-A separates complete preservation from radiation-only recovery and supplies the predecessor restore map, exact correction structure, four-Majorana control, and endpoint distinctions \cite[Theorems \texttt{thm:global}, \texttt{thm:exact}, \texttt{thm:four-majorana}, and \texttt{thm:endpoint}] {CheungHawk04A}. Those results are proved or refined below. Landmark I characterizes Born probabilities and positive-branch destructive residual rays within its declared strategy class \cite[\texttt{thm:main}]{CheungI}. En03-A specifies a minimal linear circuit interface and distinguishes that interface from assertions about a larger calculus \cite{CheungEn03A}. Here the finite fragment and its channel semantics are defined independently. Neither a field-to-qubit preparation nor the full hypotheses of Landmark I's characterization are reconstructed from its excerpt. Hawk02-E supplies the regulated quartic source--scalar architecture, finite-proxy comparison discipline, and separation of source energy from complete residual independence \cite{CheungHawk02E}. Hawk02-D concerns a different quadratic fermionic Hamiltonian and its scattering endpoint \cite{CheungHawk02D}. No quadratic survival law or scattering amplitude is spliced into the present scalar model. The apparatus and process distinctions in Em03-C, En02-B, En02-C, and En02-E are maintained \cite{CheungEm03C,CheungEn02B,CheungEn02C,CheungEn02E}. In particular, the existence of a decoder is distinct from its execution under a particular resource budget; consuming syndrome qubits is distinct from consuming the recovered quantum output. The analytic and OpenAI catalogue comparisons are confined to Appendices~\ref{app:analytic} and \ref{app:catalogue}. They are documentary separations, not premises of the channel or Hamiltonian proofs. \subsection{The successor-program correction} The phrase ``radiation recovery, then \(\Enter\) on \(R\)'' requires a correction. Applying \(\Enter\) directly to the radiation register produces a record, not an arbitrary reference-entangled diary. The correct recovery interface is \[ R\xrightarrow{\DD}C_{\mathrm{out}}, \] with quantum return. A prescribed observation may subsequently consume \(C_{\mathrm{out}}\). These two target channels are \(\id_C\) and the chosen measurement channel, respectively. Likewise, a linear program cannot silently leave a live residual wire out of its return statement. Section~\ref{sec:programs} therefore distinguishes: \begin{itemize} \item the complete program returning \(R B\); \item the local decoder program \(R\to C_{\mathrm{out}}\); \item a joint program returning \(C_{\mathrm{out}}B\); \item the observer's radiation-only marginal; \item an explicitly added readout-and-drop implementation of that marginal in a finite qubit encoding. \end{itemize} The last implementation is not inferred from the autonomous Hamiltonian. %===================================================================== \section{Complete channels, access, and distances} \label{sec:channels} Let \(C\simeq\C^d\), with \(d\ge2\), be the logical code. All output Hilbert spaces are separable. States are normal density operators, and channels are normal completely positive trace-preserving maps on trace-class operators. We abbreviate ``completely positive and trace preserving'' as CPTP. We use \[ \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1, \qquad \ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_{\diamond}. \] The diamond norm is the reference-stable trace-norm distance. For a finite input of dimension \(d\), a reference of dimension \(d\) suffices for channel discrimination; the usual finite-input conventions are as in \cite{Watrous}. For an isometry \[ V:C\longrightarrow R\otimes B, \] define \[ \VV(X)=VXV^\dagger,\qquad \NN(X)=\Tr_B\VV(X),\qquad \CC(X)=\Tr_R\VV(X). \] The inaccessible output \(B\) is \emph{complete}: it includes uncollected modes, retained source degrees of freedom, displaced carriers, preparation purifiers, and inaccessible apparatus factors. A factor is not removed from this ledger merely because an observer does not control it. For a state \(\sigma\), the replacement channel is \[ \TT_\sigma(X)=\Tr(X)\sigma. \] Set \begin{align} \erec(\NN) &=\inf_{\DD\ {\rm CPTP}}\ddia(\DD\NN,\id_C), \label{eq:erec}\\ \dleak(\CC) &=\inf_{\sigma}\ddia(\CC,\TT_\sigma). \label{eq:dleak} \end{align} The infimum in \eqref{eq:erec} is over deterministic decoders. A heralded success branch, by itself, is not such a decoder. \begin{definition}[Quantum recovery and classical observation] Quantum recovery returns a register carrying the logical code, including its correlations with an arbitrary untouched reference. A classical observation returns a state of a commutative algebra, possibly together with explicitly declared surviving quantum registers. Its target is the specified observation channel, not \(\id_C\). \end{definition} For a fixed code basis, write \(E_{ij}=\kb{i}{j}\). Every logical channel is determined by its action on all \(E_{ij}\); agreement on populations \(E_{ii}\) alone does not establish quantum preservation. \begin{lemma}[Distance conversions and stability] \label{lem:bounds} Let \(\Phi,\Psi\) have input dimension \(d\), and define \[ \ket{\Phi_d}=\frac1{\sqrt d}\sum_{i=0}^{d-1}\ket{i}_Q\ket{i}_C, \qquad J_{\mathrm n}(\Phi) =(\id_Q\otimes\Phi)(\kb{\Phi_d}{\Phi_d}). \] Then \begin{equation} \ddia(\Phi,\Psi) \le \min\left\{1,\frac d2 \norm{J_{\mathrm n}(\Phi)-J_{\mathrm n}(\Psi)}_1\right\}. \label{eq:choibound} \end{equation} For common-output isometries, \begin{equation} \ddia(\VV_1,\VV_2)\le\norm{V_1-V_2}. \label{eq:isobound} \end{equation} The same allowance holds after a common partial trace. CPTP pre- and postprocessing contract these distances. On fixed output spaces, \[ \abs{\erec(\NN_1)-\erec(\NN_2)} \le\ddia(\NN_1,\NN_2), \] and \[ \abs{\dleak(\CC_1)-\dleak(\CC_2)} \le\ddia(\CC_1,\CC_2). \] \end{lemma} \begin{proof} A normalized pure referenced input can, after an isometry on its reference support, be written as \((L\otimes I)\ket{\Phi_d}\), where \(\Tr(L^\dagger L)=d\) and \(\norm L^2\le d\). Multiplication of the normalized Choi difference by \(L\) and \(L^\dagger\) bounds its trace norm by \(d\) times the Choi trace norm. Convexity handles mixed inputs. This proves \eqref{eq:choibound}. For normalized vectors \(u,v\), \[ \dtr(\kb{u}{u},\kb{v}{v}) =\sqrt{1-\abs{\langle u,v\rangle}^2} \le\norm{u-v}. \] Apply this to referenced isometric outputs and then use convexity and partial-trace contraction. The contraction statements follow from the definition of the diamond norm. For each fixed decoder, the triangle inequality changes its error by at most the channel comparison allowance. Taking infima in both directions proves recovery stability. Distance to the replacement-channel set is likewise one-Lipschitz. \end{proof} \begin{lemma}[Local isometric dictionaries preserve the task] \label{lem:dictionaries} Separate isometric embeddings of \(R\) and \(B\) preserve \(\erec\) and \(\dleak\). They do not exchange accessible and inaccessible outputs. \end{lemma} \begin{proof} If \(J:H\to H'\) is an isometry, its channel has the CPTP retraction \[ \mathcal R_J(Y) = J^\dagger YJ+ \Tr((I-JJ^\dagger)Y)\tau_H, \] where \(\tau_H\) is any fixed state. Embedding candidate decoders or replacement states gives one inequality. Applying the retraction and contraction gives the reverse inequality. The two embeddings act on their respective local factors only. \end{proof} \begin{lemma}[Orthogonal unitary mixtures and constant-channel error] \label{lem:unitary-mixture} Suppose unitary channels \(\Ad_{U_j}\) have orthogonal Bell outputs. For real coefficients \(a_j\), \[ \frac12\left\|\sum_j a_j\Ad_{U_j}\right\|_\diamond =\frac12\sum_j\abs{a_j}. \] In particular, \begin{equation} \inf_\sigma\ddia(\id_C,\TT_\sigma) =1-d^{-2}=:a_d. \label{eq:constant-error} \end{equation} Every channel from \(C\) has recovery error at most \(a_d\). \end{lemma} \begin{proof} A Bell input gives a diagonal output difference with eigenvalues \(a_j\), proving the lower bound. Each unitary channel has diamond norm one, so the triangle inequality gives the upper bound. For any replacement state \(\sigma\), the Bell output is \(I_Q/d\otimes\sigma\). Its overlap with the target Bell projector is \(d^{-2}\). That projector therefore gives the lower bound \(1-d^{-2}\). The \(d^2\) Weyl unitaries \(X^pZ^q\) have orthogonal Bell outputs, and their uniform twirl is \(\TT_{I_C/d}\). The preceding equality gives the matching upper bound. For an arbitrary channel, the decoder which ignores its input and prepares \(I_C/d\) attains this same allowance. \end{proof} %===================================================================== \section{Complete preservation, exact correction, and no-hiding} \label{sec:preservation} \begin{theorem}[Complete unitary preservation] \label{thm:global} Let \(J:C\to H_{\mathrm{in}}\) be an isometry, let \(\ket{\Omega}\) be a fixed normalized ancillary state, and let \(U\) be unitary on the complete physical input space. Then \[ V\ket\psi=U(J\ket\psi\otimes\ket{\Omega}) \] is an isometry. Its complete channel has the CPTP left inverse \begin{equation} \mathcal L(Y) = V^\dagger YV+ \Tr((I-VV^\dagger)Y)\tau_C, \qquad \tau_C=I_C/d. \label{eq:restore} \end{equation} For every trace-class \(X\), including operators on a code--reference space, \[ \norm{(V\otimes I)X(V^\dagger\otimes I)}_1=\norm X_1. \] For the Bell input, the complete output satisfies \[ I(Q:R B)=2\log_2 d. \] \end{theorem} \begin{proof} The composition gives \(V^\dagger V=I_C\). The two terms in \eqref{eq:restore} are completely positive. Their traces sum to \(\Tr Y\), and \(\mathcal L(VXV^\dagger)=X\). An isometric embedding preserves nonzero singular values, proving the trace-norm identity. For the Bell input, the complete output is pure with \(d\) equal Schmidt coefficients. Its reduced entropies are both \(\log_2d\), and the joint entropy is zero. \end{proof} If preparation is mixed, a fixed purification is included in the complete ancillary state. If preparation depends on the diary, the code-to-prepared-state map must itself be specified; it is not covered by a fixed-ancilla assumption. The restore map is a complete-access inverse. Its existence does not imply an efficient inverse or a radiation-only inverse. It is the same global structure used in \cite[\texttt{thm:global}]{CheungHawk04A}. \begin{theorem}[Exact recovery and a constant complete complement] \label{thm:exact} For an isometry \(V:C\to R\otimes B\), the following are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item one CPTP decoder on \(R\) exactly recovers the entire code; \item \(\CC=\TT_\sigma\) for some state \(\sigma\); \item \(\CC(E_{ij})=\delta_{ij}\sigma\); \item \[ J_{\mathrm n}(\CC)=\frac{I_Q}{d}\otimes\sigma. \] \end{enumerate} When these conditions hold, there are a separable Hilbert space \(K\), a state \(\sigma_K\), and an isometry \(A:C\otimes K\to R\) such that \begin{equation} \NN(X)=A(X\otimes\sigma_K)A^\dagger. \label{eq:normalform} \end{equation} An exact decoder, defined on all trace-class operators on \(R\), is \begin{equation} \DD(Y) = \Tr_K(A^\dagger YA) +\Tr((I-AA^\dagger)Y)\tau_C. \label{eq:decoder} \end{equation} \end{theorem} \begin{proof} Linearity proves (ii) equivalent to (iii). Expansion of the Bell projector proves (iii) equivalent to (iv). Assume (ii), and write the nonzero spectral decomposition \[ \sigma=\sum_\alpha p_\alpha\kb{b_\alpha}{b_\alpha}, \qquad p_\alpha>0,\qquad \sum_\alpha p_\alpha=1. \] The index set is finite or countable. Constancy of the residual implies that every code vector is supported in \(R\otimes\supp\sigma\). Write \[ V\ket{i} = \sum_\alpha\sqrt{p_\alpha}\, \ket{r_{i\alpha}}_R\ket{b_\alpha}_B. \] The matrix-unit condition gives \[ \langle r_{j\beta},r_{i\alpha}\rangle =\delta_{ij}\delta_{\alpha\beta}. \] Hence \[ A(\ket{i}\ket{\alpha})=\ket{r_{i\alpha}} \] extends to an isometry, and \(\sigma_K=\sum_\alpha p_\alpha\kb{\alpha}{\alpha}\) gives \eqref{eq:normalform}. Both terms of \eqref{eq:decoder} are CP; their traces sum to \(\Tr Y\). The decoder recovers every \(E_{ij}\). Conversely, let \(L:R\to C\otimes G\) be a Stinespring isometry for an exact decoder. The normal Stinespring structure is the standard external dilation theorem \cite{KSW,Watrous}. Set \(Z=(L\otimes I_B)V\). For basis input \(\ket{i}\), the decoded marginal is the pure state \(\kb{i}{i}\), so \[ Z\ket{i}=\ket{i}\otimes\ket{\eta_i}. \] Exact recovery of \(E_{ij}\) requires \(\langle\eta_j,\eta_i\rangle=1\). All \(\eta_i\) are therefore one common normalized vector. Tracing the decoded register and \(G\) yields \(\CC(E_{ij})=\delta_{ij}\sigma\). \end{proof} This is the exact correction/no-hiding structure associated with \cite{KnillLaflamme,BraunsteinPati} and the normal form in \cite[\texttt{thm:exact}]{CheungHawk04A}. The local proof specifies the complete residual and the decoder's off-support completion. \begin{corollary}[The precise residual-only no-hiding conclusion] \label{cor:nohiding} If \(\NN\) is constant on the entire code, \(B\) exactly recovers that code. If \(\CC\) is constant, \(R\) exactly recovers it. For \(d\ge2\), both complementary outputs cannot independently recover the entire unknown quantum code. \end{corollary} \begin{proof} Interchange \(R\) and \(B\) in Theorem~\ref{thm:exact}. An exactly recoverable output has constant complement, which cannot recover two distinct inputs. \end{proof} \begin{remark}[Nonconstant does not mean residual-only recoverable] A nonconstant \(\CC\) means that some logical inputs are distinguished by the residual channel. It does not mean that \(B\) contains an independently recoverable copy of the entire code. Recovery may require joint access to \(R B\). The exact control in Section~\ref{sec:control} demonstrates this. \end{remark} A fixed pure residual gives \(V\ket\psi=A\ket\psi\ket{\omega}_B\). A fixed mixed residual gives the syndrome form \eqref{eq:normalform}; radiation may be mixed for a pure diary and still be exactly recoverable. %===================================================================== \section{A finite typed circuit semantics} \label{sec:semantics} \subsection{Types, ownership, and observable interfaces} Define the fragment \[ T::=\Itype\mid\Qtype\mid\Bit\mid T\otimes T \] with observable algebras \[ \mathcal A_{\Itype}=\C,\qquad \mathcal A_{\Qtype}=M_2(\C),\qquad \mathcal A_{\Bit}=\C\oplus\C. \] Hybrid states are block diagonal on their classical registers. A bit is not an unrestricted qubit under a different name. A judgement \(P:T\to T'\) consumes its input interface and returns its output interface. Sequential and tensor composition require matching interfaces. A live quantum wire is never duplicated. Every live quantum output is either returned or explicitly consumed by an admitted operation. Classical control acts blockwise. The primitive channels are \begin{align*} \New &: \Itype\to\Qtype, &\New(1)&=\kb{0}{0},\\ U &: \Qtype^{\otimes k}\to\Qtype^{\otimes k}, &\rho&\longmapsto U\rho U^\dagger,\\ \Enter &: \Qtype\to\Bit, &\rho&\longmapsto \sum_{x=0}^1\bra{x}\rho\ket{x}\kb{x}{x},\\ \Drop &: \Bit\to\Itype, &(p_0,p_1)&\longmapsto p_0+p_1. \end{align*} Return commands designate the retained interface; they are not additional dynamics. Only specified unitary constants are admitted. A semantic constant is not a claim of exact compilation over a fixed finite gate set, efficient synthesis, or physical realization. This independently defined ownership discipline agrees with the minimal interface supplied in \cite{CheungEn03A}. \subsection{Destructive partial readout} Let \(S\) be \(s\) measured qubits, and let \(T\) be the surviving quantum register. Set \[ M_x=\bra{x}_S\otimes I_T,\qquad x\in\{0,1\}^s. \] The observable instrument is \begin{equation} \mathcal E_S(\rho) = \sum_x\kb{x}{x}_K\otimes M_x\rho M_x^\dagger. \label{eq:Enter} \end{equation} The measured names in \(S\) are consumed. The surviving quantum output is \(T\), not a renamed copy of \(S\). For a pure input, \[ p_x=\norm{M_x\psi}^2,\qquad \psi_x=\frac{M_x\psi}{\sqrt{p_x}}\quad\text{if }p_x>0. \] No normalized residual is assigned to a zero-probability branch. Unnormalized branches suffice for complete-channel calculations. These formulas are the destructive Born residuals in the supplied Landmark I characterization \cite[\texttt{thm:main}]{CheungI}. The instrument is defined here directly. This does not reproduce that manuscript's uniqueness theorem without its full strategy-class hypotheses. \begin{theorem}[Reference-preserving adequacy] \label{thm:adequacy} For every finite acyclic instruction tree, and each fixed classical input block: \begin{enumerate}[label=\textup{(\roman*)}] \item the denotation is a CPTP channel to the declared hybrid output; \item there are quantum path maps \(K_h\) satisfying \[ \sum_hK_h^\dagger K_h=I; \] \item each reported output block is the sum of its unnormalized maps \(X\mapsto K_hXK_h^\dagger\); \item the same denotation holds with an arbitrary untouched quantum reference; \item every evaluation branch terminates within the finite height of the tree. \end{enumerate} \end{theorem} \begin{proof} Allocation is an isometry, and a unitary gate is trace preserving. Destructive readout satisfies \(\sum_xM_x^\dagger M_x=I\). Classical control selects a complete channel on each classical block. Classical discard sums blocks without conditional normalization. Induction composes path maps. Backward induction, using completeness at each readout node, gives \(\sum_hK_h^\dagger K_h=I\). Grouping histories with the same reported output gives the corresponding hybrid block. Tensoring each quantum path map with the reference identity proves complete positivity and the reference statement. A finite acyclic tree has a finite maximum branch length. \end{proof} The theorem is about the declared finite fragment. It is neither a complexity theorem nor a normalization theorem for an unprovided larger calculus. It uses the complete branch accounting emphasized in \cite{CheungEn02C}. \begin{proposition}[Finite-dimensional semantic realization] \label{prop:finite-realization} A finite-dimensional isometry has a qubit encoding and a unitary extension after pure allocation. A finite-dimensional CPTP channel can be represented by pure allocation, a specified unitary, destructive readout of its unused work register, discard of those work outcomes, and return of its quantum output. \end{proposition} \begin{proof} Choose orthonormal basis dictionaries into sufficiently large qubit registers. Complete the columns of the encoded isometry to an orthonormal basis, obtaining a unitary extension. For Kraus operators \(D_j\), the map \[ L\ket\psi=\sum_jD_j\ket\psi\ket{j}_G \] is an isometry because \(\sum_jD_j^\dagger D_j=I\). A unitary extension implements this coherent map. Readout of \(G\) and discard of the outcome give \[ X\longmapsto\sum_jD_jXD_j^\dagger. \] The designated logical output remains quantum. \end{proof} When a logical dimension is not a power of two, the logical space is an encoded subspace of its qubit register. Every physical channel is completed on invalid padded inputs, for example by fixed-state preparation. Logical recovery is tested on \(C\), including references; it is not an assertion about arbitrary invalid padded inputs. An infinite-dimensional decoder or syndrome space need not admit a finite program. Theorem~\ref{thm:exact} still supplies a normal CPTP decoder under its hypotheses. %===================================================================== \section{Explicit programs and their output ledgers} \label{sec:programs} For a finite proxy, let \(Q_C,Q_R,Q_B\) denote qubit dictionaries for the logical code and the two outputs. These are register abbreviations, with encoded-subspace conditions understood. Let \(U_{\mathrm{body},t}\) be a specified padded unitary whose restriction implements the prepared proxy isometry. \subsection{Complete coherent preservation} \label{subsec:coherent-program} The complete program is \[ P_{\mathrm{coh},t}:Q_C\longrightarrow Q_R\otimes Q_B. \] Its body is \begin{verbatim} a := new; (C,a) := EncodeAndPrepare(C,a); (R,B) := Evolve_t(C,a); returnQ(R,B); \end{verbatim} Here \texttt{new} abbreviates the required finite pure allocation. The preparation and evolution are specified unitary extensions. Together they realize \(U_{\mathrm{body},t}\) on the encoded input. There is no \(\Enter\) in this coherent body. Its logical denotation is \(\widehat\VV_t\). The global restore map is \eqref{eq:restore}; in finite dimension it is realizable by Proposition~\ref{prop:finite-realization}. This is the typed reading of complete preservation in \cite{CheungHawk04A,CheungEn03A}. The program begins after the carrier dictionary and preparation map have been defined. It constructs neither a quantum field nor a metric. \subsection{A local quantum decoder} \label{subsec:decoder-program} For a specified finite CPTP decoder, the local program is \[ P_{\DD}:Q_R\longrightarrow Q_C^{\mathrm{out}}, \] with body \begin{verbatim} w := new; (Cout,G) := DecoderUnitary(R,w); y := Enter(G); dropBit(y); returnQ(Cout); \end{verbatim} All unused work wires are included in \(G\). The program consumes syndrome/work qubits, not the diary output. This interface is consistent with the quantum-return decoder programs in \cite{CheungEn02B}. Applying this program locally to the coherent output gives \[ P_{\mathrm{keep},t} = (P_{\DD}\otimes\id_{Q_B})P_{\mathrm{coh},t} : Q_C\longrightarrow Q_C^{\mathrm{out}}\otimes Q_B. \] Its body returns \texttt{(Cout,B)}. The observer who controls only the first returned factor sees \[ \DD\widehat\NN_t. \] Thus radiation-only access is a declared marginal of a complete output, not an implicit rule deleting a live quantum wire. \subsection{A strict one-register observable program} \label{subsec:strict-program} If one additionally wants a finite program whose \emph{observable} output is only \(Q_C^{\mathrm{out}}\), the residual must be consumed explicitly: \begin{verbatim} (R,B) := CoherentBody_t(C); x := Enter(B); dropBit(x); Cout := Decode(R); returnQ(Cout); \end{verbatim} Its channel is again \(\DD\widehat\NN_t\), by Theorem~\ref{thm:drop} below. This readout-and-drop is an external interface implementation. It is not a claim that the autonomous Hamiltonian measures \(B\), nor that the omitted physical apparatus has disappeared. The complete dilation retains its apparatus outputs. \subsection{Direct radiation observation and observation after recovery} \label{subsec:observation-program} Directly measuring \(R\) denotes a channel \[ \MM_R\NN_t:C\longrightarrow K, \] not a quantum decoder. For observation \emph{after} recovery, let \(F\) be a specified basis-selection unitary on the encoded output and append \begin{verbatim} Cout := F(Cout); x := Enter(Cout); returnBit(x); \end{verbatim} The observable channel is \(\MM_F\DD\NN_t\). Its intended target is \(\MM_F\), not \(\id_C\). For non-power-of-two codes, \(\MM_F\) includes the declared padded implementation and logical restriction. \begin{proposition}[Recovery certifies subsequent records] \label{prop:records} If \[ \ddia(\DD\NN_t,\id_C)\le\varepsilon, \] then every prescribed terminal observation satisfies \[ \ddia(\MM_F\DD\NN_t,\MM_F)\le\varepsilon. \] The converse for one chosen observation is false. \end{proposition} \begin{proof} The upper bound is CPTP postprocessing contraction. Computational dephasing preserves the computational-basis observation exactly but annihilates off-diagonal matrix units. That observation equality does not imply quantum recovery. \end{proof} Evaluating pre-written coherent code and executing the particular command \(\Enter\) are therefore different operations. A coherent finite program may terminate without any destructive readout. %===================================================================== \section{Classical records cannot replace an unknown quantum diary} \label{sec:classical} A measure-and-prepare channel has the form \[ \mathcal E(X)=\sum_x\Tr(F_xX)\sigma_x, \] where \(\{F_x\}\) is a POVM and the \(\sigma_x\) are states. Such channels are entanglement breaking \cite{HorodeckiShorRuskai}. The following proof gives the quantitative obstruction needed here. \begin{theorem}[Classical-output and measure-and-prepare obstruction] \label{thm:classical} Let \(\mathcal E\) be a measure-and-prepare channel from \(C\) to an arbitrary output, and let \(\DD\) be any CPTP decoder to \(C\). Then \begin{equation} \ddia(\DD\mathcal E,\id_C)\ge1-d^{-1}. \label{eq:classical-obstruction} \end{equation} In particular, the bound applies to every finite adaptive process whose only retained output is classical. Computational readout followed by basis-state preparation attains the bound. \end{theorem} \begin{proof} The Bell-input output of \(\DD\mathcal E\) is separable across the reference--output partition. For a normalized product vector, \[ \abs{\langle\Phi_d,u\otimes v\rangle}^2\le\frac1d \] by Cauchy--Schwarz. Convexity gives Bell overlap at most \(1/d\) for every separable state. Testing the Bell projector proves \eqref{eq:classical-obstruction}. A channel to a finite classical register has the form \[ X\longmapsto\sum_x\Tr(F_xX)\kb{x}{x}. \] A decoder from that register prepares a state for each outcome. Adaptive processing before the final classical output does not change this form. For attainability, define \[ Z=\sum_{j=0}^{d-1}e^{2\pi\ii j/d}\kb{j}{j}. \] Computational dephasing is \[ \Delta_d=\frac1d\sum_{k=0}^{d-1}\Ad_{Z^k}. \] Its \(d\) Bell outputs are orthogonal. Lemma~\ref{lem:unitary-mixture} gives \(\ddia(\Delta_d,\id_C)=1-1/d\). \end{proof} For a qubit, every record-only reconstruction has error at least \(1/2\). The hypothesis is that only the classical register is retained: a surviving quantum memory, or quantum side information supplied to the receiver, is a different access task. A classical record may identify a basis label or support repeated-preparation tomography. It does not preserve an arbitrary unknown quantum state and its reference correlations in one run. %===================================================================== \section{Residual readout, local forgetting, and the apparatus} \label{sec:discard} \subsection{The explicitly added residual programs} For a finite encoded residual, define \[ P_{\mathrm{readB}}:Q_R\otimes Q_B\to Q_R\otimes\Bit^{\otimes b} \] by \begin{verbatim} x := Enter(B); return(R,x); \end{verbatim} and define \[ P_{\mathrm{forgetB}}:Q_R\otimes Q_B\to Q_R \] by \begin{verbatim} x := Enter(B); dropBit(x); returnQ(R); \end{verbatim} Neither program follows merely from the autonomous Hamiltonian. The derived notation \[ \ForgetQ=\Drop\circ\Enter \] denotes this finite destructive-readout-and-forget implementation; it is not a primitive assertion of physical annihilation. \begin{theorem}[Readout-and-drop is exactly local partial trace] \label{thm:drop} For every joint state, including any external reference, \[ \sum_x (\bra{x}_B\otimes I)\rho(\ket{x}_B\otimes I) =\Tr_B\rho. \] Consequently, \begin{equation} P_{\mathrm{forgetB}}\VV_t=\NN_t. \label{eq:forget-equality} \end{equation} Its radiation-only recovery error is exactly the pre-existing \(\erec(\NN_t)\). More generally, every trace-preserving operation on \(B\), with all its outputs ignored, leaves the state on the untouched reference and \(R\) unchanged. \end{theorem} \begin{proof} Partial trace in the computational basis is the sum of the diagonal \(B\)-blocks, which proves the first equality and \eqref{eq:forget-equality}. For the general statement, let \(\Lambda_B\) be the local channel. Its Heisenberg adjoint is unital. For every bounded observable \(A\) on the untouched factors, \[ (\id\otimes\Lambda_B^*)(A\otimes I)=A\otimes I. \] All expectations on those factors are unchanged. This argument applies to normal channels on separable spaces; equivalently, in a Kraus representation, \(\sum_jL_j^\dagger L_j=I_B\). \end{proof} This is no-signalling, not a statement that conditional radiation states cannot change. Conditioning on a supplied measurement outcome is a different operation. For an infinite residual, partial trace remains well defined. The theorem does not assert a finite program that measures infinitely many modes. \begin{proposition}[Classical assistance is a different access task] \label{prop:assistance} Suppose residual readout produces the hybrid channel \[ \mathcal N_{RK}(X) = \sum_x\kb{x}{x}\otimes\mathcal N_x(X), \qquad \sum_x\mathcal N_x=\NN. \] A deterministic decoder on its classical interface has the form \[ \mathcal D_{RK}(\mathcal N_{RK}(X)) = \sum_x\mathcal D_x(\mathcal N_x(X)), \] where each \(\mathcal D_x\) is CPTP on \(R\). Moreover, \[ \erec(\mathcal N_{RK})\le\erec(\NN). \] The inequality can be strict. \end{proposition} \begin{proof} A channel on a direct sum of classical blocks acts by a CPTP channel on each block and adds the outputs. A decoder can ignore \(K\), so every radiation-only strategy is available with \(R K\) access. Example~\ref{ex:assistance} proves strictness. \end{proof} \subsection{The coherent apparatus ledger} \begin{proposition}[A complete readout dilation retains omitted information] \label{prop:measurement-dilation} For the destructive instrument \eqref{eq:Enter}, define \begin{equation} W_{\mathrm{read}}\ket\psi = \sum_x\ket{x}_K\ket{x}_E\otimes M_x\ket\psi. \label{eq:measurement-isometry} \end{equation} This is an isometry. Tracing \(E\) gives the observable hybrid readout instrument. The complete output \(K E T\) preserves the input globally, whereas the classical interface \(K\), by itself, generally does not. \end{proposition} \begin{proof} The identities \(\langle x,y\rangle=\delta_{xy}\) and \(\sum_xM_x^\dagger M_x=I\) give \(W_{\mathrm{read}}^\dagger W_{\mathrm{read}}=I\). Tracing the orthogonal \(E\)-labels removes cross-outcome terms and yields \eqref{eq:Enter}. Complete recoverability follows from Theorem~\ref{thm:global}. \end{proof} The physical carrier \(K\) in the coherent dilation is a quantum system. It becomes the observable \(\Bit\) interface only after the inaccessible factor \(E\) is omitted. A coherent superposition across outcome labels is not a state of the classical algebra. For a fully measured qubit, \[ a\ket0+b\ket1 \longmapsto a\ket0_K\ket0_E+b\ket1_K\ket1_E. \] Neither marginal retains the unknown phase; joint \(K E\) access does. There is no conflict with no-hiding, because neither marginal channel is constant on the entire qubit code. Classical discard may be dilated by moving the record carrier into another inaccessible register. Under complete unitary implementation, that register and the other apparatus factors remain in the ledger. The observable trace does not prove physical disappearance. This accounting is compatible with the joint-apparatus and coherent comparison discipline in \cite{CheungEm03C,CheungEn02C}. No apparatus model from those manuscripts is silently substituted for the dilation specified here. \subsection{Two exact destination tests} \begin{example}[The whole diary is initially in the residual] \label{ex:bin} For \[ V\ket\psi=\ket0_R\ket\psi_B, \] radiation is constant and the complete residual exactly recovers the diary. Computational readout of \(B\) retains only basis populations; its bits cannot replace the diary by Theorem~\ref{thm:classical}. Readout-and-drop leaves radiation constant. A complete unitary measurement dilation remains globally reversible. \end{example} \begin{example}[A diary-independent bit can assist recovery] \label{ex:assistance} For a qubit, consider \[ V\ket\psi= \frac1{\sqrt2} \left(\ket\psi_R\ket0_B+Z\ket\psi_R\ket1_B\right). \] The radiation channel is \(\Delta_Z=(\id+\Ad_Z)/2\), with optimal recovery error \(1/2\). Computational readout of \(B\) gives a uniform bit \(x\). Supplying \(x\) to the observer and applying \(Z^x\) to \(R\) gives exact quantum recovery. The bit alone contains no diary information. \end{example} \begin{proof}[Verification of Example~\ref{ex:assistance}] The unnormalized branch is \[ \rho\longmapsto\frac12 Z^x\rho Z^x. \] Correction by \(Z^x\), followed by summation, returns \(\rho\). Without correction, summation returns \(\Delta_Z(\rho)\). Each outcome has probability \(1/2\), independently of the input. The two inputs \(\ket+\) and \(\ket-\) have identical dephased outputs. If a decoder has error \(\varepsilon\), the triangle inequality and contraction imply \(1\le2\varepsilon\). The identity decoder attains \(1/2\) by Lemma~\ref{lem:unitary-mixture}. \end{proof} This example separates classical side information supporting a quantum correction from a classical replacement for the quantum diary. %===================================================================== \section{An independently specified regulated microscopic model} \label{sec:model} The programming semantics does not construct the Hamiltonian below. The model follows the regulated quartic source--scalar architecture in \cite{CheungHawk02E,CheungHawk04A}. We use units \(\hbar=1\). \subsection{Frozen source and finite-band reservoir} Let \(N=2n\ge4\), and let \[ \chi_i^\dagger=\chi_i,\qquad \{\chi_i,\chi_j\}=2\delta_{ij}I \] act on \(\mathcal H_S=\FF(\C^n)\). For frozen real coefficients \(J_{ijkl}\), set \[ H_{\mathrm{bare}} =\sum_{i0\) and \(0