% Em04-A-v2.tex % Standalone pdfLaTeX manuscript; no external figures or bibliography files. % % Suggested automated build: % latexmk -norc -pdf -Werror -halt-on-error -file-line-error \ % -interaction=nonstopmode -outdir=build Em04-A-v2.tex % % Direct alternative: run pdfLaTeX until references stabilize. % % No compilation, solver execution, validated numerical run, independent % symbolic execution, or machine-checked proof is claimed in this source. % % Provenance correction: % The supplied verification plan reports an originating simulator export. % Its described conventions are recorded as reported correspondences, not % as a completed inspection of a line-numbered, archived source export. % % Principal changes from v1: % - separate mathematical proofs from implementation and execution evidence; % - give the complete partial-transpose spectrum with zero multiplicities; % - state the finite-chi hypothesis explicitly; % - replace the main midpoint rule by the composite trapezoidal rule; % - account for arithmetic errors and loss of positivity of rounded matrices; % - strengthen the completed-POVM total-variation bound; % - provide cellwise residual and physicality-repair certificates; % - move unrelated programme bookkeeping to an appendix. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,tabularx,longtable,array} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \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{\dd}{\,\mathrm d} \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{\gr}{\gamma_{\mathrm r}} \newcommand{\gd}{\gamma_{\mathrm d}} \newcommand{\Diss}{\mathcal D} \newcommand{\Gen}{\mathcal L} \newcommand{\Hil}{\mathcal H} \newcommand{\Vac}{\Omega} \newcommand{\Nexc}{\widehat N_{\mathrm{exc}}} \newcommand{\trdist}{d_{\mathrm{tr}}} \newcommand{\tv}{d_{\mathrm{TV}}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\spec}{spec} \DeclareMathOperator{\vecop}{vec} \DeclareMathOperator{\RePart}{Re} \title{\textbf{Em04-A}\\[0.5em] Exact Reduction, Two-Coordinate Scaling, and Transient Entanglement in a Locally Dissipative XY Chain\\[0.4em] \large Local Instruments and Validated Approximation Interfaces} \author{} \date{Version 2} \begin{document} \maketitle \begin{abstract} We specify a finite open XY chain with uniform zero-temperature relaxation and local dephasing. Its evolution from a single excitation admits an exact vacuum-plus-single-excitation reduction to an \(N\times N\) density matrix and a scalar survival probability. We derive explicit concurrence, logarithmic-negativity, purity, and computational-basis-coherence formulas, including the complete partial-transpose spectrum. At fixed coherent time and fixed dimensionless detunings, two dissipative coordinates \((\chi,r)\) are required: excitation survival and a short-time concurrence expansion disprove single-\(\chi\) universality. We prove entanglement generation, dissipative decay, conditional mixing, and controlled finite-window limits. Separate results establish finite local instrument consistency and typed deferred readout. For an independently specified oscillator, we prove compressed-position POVM identities, state-preparation and tail bounds, a composite-trapezoidal error estimate, and an arithmetic-aware completion to a valid POVM. A cellwise residual theorem provides a validated state-error interface without assuming positivity of the approximation. Mathematical proofs are distinguished from source alignment, numerical regression, validated numerical certificates, and document construction. No execution or completed verification package is asserted. \end{abstract} \section{Scope, evidence categories, and model specification} \label{sec:contract} \subsection{Scope} This is an independent mathematical construction under specified quantum semantics. Density operators, completely positive channels, Lindblad dynamics, and Born probabilities are assumptions, not conclusions derived from a measurement-strategy characterization. No earlier landmark manuscript is a formal premise of the proofs. The spin-chain and oscillator constructions are distinct models. Neither is identified with a gravitational mediator, quantum field, microscopic evaporation channel, or characteristic-product coefficient. Documentary programme restrictions are recorded separately in Appendix~\ref{app:scope}. \subsection{Evidence categories and present status} \label{subsec:evidence} We distinguish: \begin{description}[style=nextline,leftmargin=0pt] \item[Analytical proof.] A theorem established by an explicit argument under stated hypotheses. \item[Source alignment.] A documented comparison with an identified implementation, including source locations and conventions. \item[Numerical regression.] Reproducible floating-point comparisons with predeclared tolerances. \item[Validated numerical certificate.] Outward-rounded enclosures and a finite error budget for a specified calculation. \item[Document construction.] Successful compilation, resolved references, and inspection of the PDF. \end{description} The theorem and proof environments below belong to the first category. A proved error-bound theorem is an \emph{interface for} a numerical certificate; it is not evidence that its hypotheses have been numerically enclosed for a particular run. The supplied verification plan reports that an originating simulator export was supplied as documentary material and describes its conventions. These reported correspondences are recorded in Appendix~\ref{app:alignment}. This revision does not include an archived, line-numbered copy of that export or a completed source-inspection record. Source alignment therefore remains to be independently established; it is not replaced by the assertion that code was unavailable, nor by an unsupported assertion that inspection passed. No execution archive, compiler log, independent symbolic-check output, validated numerical certificate, or machine-checked proof accompanies this revision. The execution-dependent release gates in Appendix~\ref{app:verification} remain unverified. \subsection{Notation} For matrices, \[ \norm X_1=\Tr\sqrt{X^\dagger X},\qquad \norm X_{\mathrm F}=\sqrt{\Tr X^\dagger X},\qquad \norm X_\infty=\text{operator norm}. \] For density operators, \[ \trdist(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1. \] All entanglement logarithms are base two. A density operator is positive semidefinite and has trace one. \begin{definition}[Specified zero-temperature chain] \label{def:model} Let \(N\ge2\), \(J>0\), \(\gr,\gd\ge0\), and \(\epsilon_k\in\R\), \(0\le k0\), and substituting \(R=qS\) gives \eqref{eq:S}. This conditional equation is itself GKSL: \[ 2\gd(\Delta S-S) =\sum_k\Diss[\sqrt{2\gd}\,\kb kk](S). \] Thus it has a unique physical solution. The reconstructed state satisfies the full equation and initial condition, so uniqueness proves the identity. \end{proof} \begin{remark}[Exactness and exclusions] The physical tensor factorization remains that of \(N\) sites. The \(N\)-level matrix \(S\) is a computational reduction, not a new factorization for evaluating entanglement. Uniform relaxation is a hypothesis of this scalar-survival theorem. Nonuniform relaxation still preserves the vacuum-plus-one-excitation block class, but generally replaces \(-\gr R\) by \(-\{D,R\}/2\), with \(D\) the diagonal rate matrix; the universal scalar factorization above then need not hold. Bath raising, higher-excitation preparations, and vacuum--excitation coherences are not covered by the theorem. \end{remark} \section{Exact diagnostic and entanglement identities} \label{sec:diagnostics} \subsection{Two-site concurrence} For a two-qubit density matrix \(\omega\), let \[ \widetilde\omega=(Y\otimes Y)\overline\omega(Y\otimes Y). \] The concurrence is \[ C(\omega)=\max\{0,\lambda_1-\lambda_2-\lambda_3-\lambda_4\}, \] where the decreasing \(\lambda_\ell\) are the nonnegative square roots of the eigenvalues of \(\omega\widetilde\omega\) \cite{Wootters}. \begin{theorem}[Exact pairwise concurrence] \label{thm:concurrence} For distinct sites \(i,j\), in the basis \((\ket{00},\ket{01},\ket{10},\ket{11})\), \[ \rho_{ij}= \begin{pmatrix} 1-R_{ii}-R_{jj}&0&0&0\\ 0&R_{jj}&R_{ji}&0\\ 0&R_{ij}&R_{ii}&0\\ 0&0&0&0 \end{pmatrix}. \] Consequently \begin{equation} \boxed{C_{ij}=2|R_{ij}|=2q|S_{ij}|,} \label{eq:C} \end{equation} and \begin{equation} C_{ij}\le2\sqrt{R_{ii}R_{jj}} \le R_{ii}+R_{jj}\le q. \label{eq:Cbound} \end{equation} \end{theorem} \begin{proof} An excitation outside the pair reduces to \(\ket{00}\). Only coherences supported entirely in the pair survive the partial trace, giving the displayed matrix. Set \(a=R_{jj}\), \(b=R_{ii}\), and \(z=R_{ji}\). Direct multiplication with the spin-flipped matrix gives square-root eigenvalues \[ \sqrt{ab}+|z|,\quad\sqrt{ab}-|z|,\quad0,\quad0. \] Positivity implies \(|z|\le\sqrt{ab}\). Substitution yields \(C=2|z|\), and the same positivity inequality followed by the arithmetic--geometric mean inequality proves the bounds. \end{proof} \subsection{Complete partial-transpose spectrum} For a partition \(A\sqcup B=\{0,\ldots,N-1\}\) with both parts nonempty, let \(R_{AB}\) have rows in \(A\), columns in \(B\). Use \[ \mathcal N^{A|B}(\rho) =\frac{\norm{\rho^{T_B}}_1-1}{2},\qquad E_{\mathcal N}^{A|B}(\rho)=\log_2\norm{\rho^{T_B}}_1, \] with computational-basis partial transpose \cite{VidalWerner}. \begin{theorem}[Spectrum, negativity, and separability] \label{thm:negativity} Let \(p_0=1-q\) and \(s=\norm{R_{AB}}_{\mathrm F}\). As a multiset, the complete spectrum is \begin{equation} \spec(\rho^{T_B}) = \spec(R_{AA})\uplus\spec(R_{BB}^{\mathsf T}) \uplus\{\lambda_+,\lambda_-\} \uplus\{0\}^{\,2^N-N-2}, \label{eq:PTspec} \end{equation} where \[ \lambda_\pm=\frac{p_0\pm\sqrt{p_0^2+4s^2}}{2}. \] Zeros already present in the first three multisets are retained. The formula holds for \(s=0\) as well. Therefore \begin{align} \mathcal N^{A|B}(\rho) &=\frac{\sqrt{p_0^2+4\norm{R_{AB}}_{\mathrm F}^2}-p_0}{2}, \label{eq:neg}\\ \boxed{ E_{\mathcal N}^{A|B}(\rho) =\log_2\!\left( 1+\sqrt{p_0^2+4\norm{R_{AB}}_{\mathrm F}^2}-p_0 \right). } \label{eq:EN} \end{align} Within the state class \eqref{eq:reduction}, \[ \rho\text{ is separable across }A|B \quad\Longleftrightarrow\quad R_{AB}=0. \] Also, \begin{equation} E_{\mathcal N}^{A|B}(\rho) \le\log_2\!\left(q+\sqrt{(1-q)^2+q^2}\right) \le\log_2(1+q). \label{eq:ENbound} \end{equation} \end{theorem} \begin{proof} A cross matrix unit transforms according to \[ \kb{e_i^A,0_B}{0_A,e_j^B} \ \longmapsto\ \kb{e_i^A,e_j^B}{0_A,0_B}. \] The vacuum and the states with one excitation on each side therefore carry the block \[ \begin{pmatrix}p_0&v^\dagger\\v&0\end{pmatrix}, \qquad v=\sum_{i\in A,j\in B}R_{ij}\ket{e_i^A,e_j^B}, \qquad \norm v=s. \] The cross-sector basis is orthonormal. For \(s>0\), restriction to the vacuum and \(v/s\) gives \(\lambda_\pm\); the remaining directions give zeros. For \(s=0\), choose any one cross-sector basis vector instead, giving eigenvalues \(p_0,0\). Such a vector exists because both parts are nonempty. The other nonzero blocks are \(R_{AA}\) and \(R_{BB}^{\mathsf T}\), on mutually orthogonal local single-excitation spaces. All remaining sectors are zero. Their dimensions give exactly \(2^N-N-2\) additional zeros in \eqref{eq:PTspec}. The positive local blocks have combined trace \(q\), giving the trace norm and the negativity formulas. If \(R_{AB}\ne0\), then \(\lambda_-<0\). Every separable state has positive partial transpose, by transposing the positive second factor in each product term of a separable decomposition. Hence this state is entangled. If \(R_{AB}=0\), the vacuum term and the two local excitation blocks are positive product terms. Normalizing the nonzero terms by their traces gives a separable decomposition. Finally, \[ \norm{R_{AB}}_{\mathrm F}^2 \le\sum_{i\in A,j\in B}R_{ii}R_{jj} \le\frac{q^2}{4}. \] Insert this into \eqref{eq:EN}. The inequality \(\sqrt{(1-q)^2+q^2}\le1\) proves the last bound. \end{proof} \subsection{Purity and computational-basis coherence} Use the specified normalized diagnostic \[ C_{\ell_1}^{\mathrm{norm}}(\rho) =\frac{\sum_{x\ne y}|\rho_{xy}|}{2^N-1}. \] Its numerator is the usual basis-dependent \(\ell_1\) coherence \cite{BCP}; the denominator is an additional convention. \begin{corollary}[Purity, coherence, and vacuum convergence] \label{cor:diagnostics} For \eqref{eq:reduction}, \begin{align} \Tr\rho^2&=(1-q)^2+q^2\Tr S^2, \label{eq:purity}\\ C_{\ell_1}^{\mathrm{norm}}(\rho) &=\frac{q\sum_{i\ne j}|S_{ij}|}{2^N-1} \le\frac{N-1}{2^N-1}q, \label{eq:coherence}\\ \frac{\mathrm d}{\mathrm dt}\Tr S^2 &=-4\gd\norm{S-\Delta S}_{\mathrm F}^2\le0. \label{eq:Spurity} \end{align} For \(\gr>0\), \[ \trdist(\rho(t),\kb{\Vac}{\Vac})=q(t),\qquad \Tr\rho(t)^2\longrightarrow1. \] \end{corollary} \begin{proof} The vacuum and excitation blocks are orthogonal, proving \eqref{eq:purity}. All full-state off-diagonal entries belong to \(qS\). Positivity and Cauchy--Schwarz give \[ \sum_{i\ne j}|S_{ij}| \le\left(\sum_i\sqrt{S_{ii}}\right)^2-1\le N-1. \] Differentiating conditional purity eliminates the commutator, and \(\Tr(S(S-\Delta S))=\norm{S-\Delta S}_{\mathrm F}^2\). The difference from the vacuum has blocks \(-q\) and \(qS\), so its trace norm is \(2q\). The limiting purity follows from \eqref{eq:purity}. \end{proof} \begin{remark}[Nonmonotone full purity] For the default pure preparation with \(\gd=0\), \(S\) remains pure and \(\Tr\rho^2=(1-q)^2+q^2\). At positive relaxation this decreases to \(1/2\) at \(t=(\log2)/\gr\) and subsequently increases to one. At fixed \(t>0\), it is also nonmonotone as a function of \(\gr\). Vacuum convergence does not prove monotonicity in relaxation rate. The normalization in \eqref{eq:coherence} must likewise be separated from any claimed chain-length dynamical effect. \end{remark} \section{Scaling and short-time parameter dependence} \label{sec:scaling} \begin{definition}[Protocols and coordinates] Protocol A fixes physical observation times \(t\). Protocol B fixes coherent observation times \(\tau=Jt\). Let \[ \Gamma_2=\gd+\frac{\gr}{2}. \] When \(\Gamma_2>0\), define \begin{equation} \chi=\frac{J}{\Gamma_2}>0,\qquad r=\frac{\gr}{2\Gamma_2}\in[0,1]. \label{eq:coordinates} \end{equation} Equivalently, \[ \gr=\frac{2rJ}{\chi},\qquad \gd=\frac{(1-r)J}{\chi}. \] Here \(\Gamma_2\) is an isolated qubit's transverse decay rate under the specified convention, not the intersite dephasing rate. \end{definition} \begin{theorem}[Exact two-coordinate scaling] \label{thm:scaling} With \(\widetilde h=h/J\), Protocol-B dynamics satisfy \begin{align} S'(\tau)&=-\ii[\widetilde h,S] -\frac{2(1-r)}{\chi}(S-\Delta S), \label{eq:scaledS}\\ q(\tau)&=e^{-2r\tau/\chi}. \label{eq:scaledq} \end{align} After fixing \(N,S_0,\epsilon_i/J\), the pair \((\chi,r)\) determines these trajectories. A common detuning shift has no effect. Multiplying \(J,\epsilon_i,\gr,\gd\) by a common \(c>0\) replaces the physical-time trajectory \(\rho(t)\) by \(\rho(ct)\) and leaves the corresponding Protocol-B trajectory unchanged. Nevertheless two models with the same finite \(\chi\) and different \(r\) cannot have identical full Protocol-B trajectories. \end{theorem} \begin{proof} Substitution proves the scaled equations. A common detuning shift changes \(\widetilde h\) by a scalar identity, which disappears from the commutator. Simultaneous scaling changes the full generator to \(c\Gen\), because collapse amplitudes scale by \(\sqrt c\). Finally, the excitation expectation is \(e^{-2r\tau/\chi}\), which differs for different \(r\) at every \(\tau>0\). \end{proof} \begin{remark}[Zero dissipation] If \(\Gamma_2=0\), both rates vanish. The model is unitary, and \((\chi,r)\) is not a finite coordinate chart. Limits \(\chi\to\infty\) at fixed \(r\) are meaningful, but the exactly nondissipative model has no distinguished \(r\). \end{remark} \begin{theorem}[Short-time concurrence] \label{thm:shorttime} For the default localized preparation and \(\Gamma_2>0\), \begin{equation} C_{01}(\tau) =2\tau-\frac{2(1+r)}{\chi}\tau^2+O(\tau^3), \qquad \tau\downarrow0. \label{eq:shortC} \end{equation} The formula holds for arbitrary fixed real detunings and includes \(r=0,1\). In physical time, for all nonnegative rates, it reads \[ C_{01}(t)=2Jt-2J(\gr+\gd)t^2+O(t^3). \] \end{theorem} \begin{proof} Set \(P=\kb00\), \(d_i=-\epsilon_i/J\), and \(\kappa=2(1-r)/\chi\). For \(\mathcal A(M)=-\ii[\widetilde h,M]-\kappa(M-\Delta M)\), analyticity gives \[ S=P+\tau\mathcal A(P)+\frac{\tau^2}{2}\mathcal A^2(P) +O(\tau^3). \] Direct matrix multiplication on the open chain yields \[ \mathcal A(P)_{01}=\ii,\qquad -[\widetilde h,[\widetilde h,P]]_{01}=d_0-d_1, \qquad \mathcal A^2(P)_{01}=d_0-d_1-\ii\kappa. \] Thus \[ S_{01}=\ii\tau+\frac{d_0-d_1-\ii\kappa}{2}\tau^2+O(\tau^3), \qquad |S_{01}|=\tau-\frac{\kappa}{2}\tau^2+O(\tau^3). \] The detuning term affects the modulus only at third order. Multiplying by \(2q\), with \(q=1-(2r/\chi)\tau+O(\tau^2)\), proves \eqref{eq:shortC}. The same calculation with physical coefficients proves the last formula, including zero dissipation. All remainders concern a fixed model, not unbounded detuning families. \end{proof} The survival law and \eqref{eq:shortC} independently establish the failure of general single-\(\chi\) universality. \section{Transient entanglement and controlled comparisons} \label{sec:limits} \begin{theorem}[Generation and dissipative decay] \label{thm:transient} For the default localized preparation, finite rates, and finite detunings, \(C_{01}>0\) on a sufficiently short positive-time interval. Every cut separating sites \(0,1\) is entangled there. If \(\gr>0\), every pairwise concurrence and bipartite logarithmic negativity tends to zero. The concurrence \(C_{01}\) attains a strictly positive global maximum at finite positive time. \end{theorem} \begin{proof} The leading coefficient in the physical-time expansion is \(2J>0\). A cut separating the two sites contains the nonzero coherence in \(R_{AB}\), so Theorem~\ref{thm:negativity} applies. For \(\gr>0\), \(q\to0\), and \eqref{eq:Cbound}, \eqref{eq:ENbound} prove decay. A continuous function vanishing at zero and infinity but positive somewhere attains a positive maximum on a sufficiently large compact time interval, which is also its global maximum. \end{proof} \begin{theorem}[Conditional mixing with dephasing] \label{thm:mixing} If \(J>0\) and \(\gd>0\), then \[ S(t)\longrightarrow\frac{\Id_N}{N}. \] For the fixed model there are \(M,g>0\) such that \[ \norm{S(t)-\Id_N/N}_{\mathrm F}\le Me^{-gt}. \] Consequently when \(\gr=0,\gd>0\), all the entanglement diagnostics above tend to zero, and the full limiting state is the separable state \[ \frac1N\sum_k\kb{e_k}{e_k}. \] \end{theorem} \begin{proof} On all complex matrices, the conditional generator \(\mathcal A(M)=-\ii[h,M]-2\gd(M-\Delta M)\) obeys \[ \RePart\Tr(M^\dagger\mathcal A(M)) =-2\gd\norm{M-\Delta M}_{\mathrm F}^2. \] If \(\mathcal A(M)=\lambda M\), \(M\ne0\), and \(\RePart\lambda=0\), then \(M\) is diagonal. The diagonal equation gives \(\lambda M_{ii}=0\), hence \(\lambda=0\). The neighboring off-diagonal equations give \(J(M_{i+1,i+1}-M_{ii})=0\). Connectedness implies \(M=c\Id_N\). The invariant trace-zero subspace therefore has its finite spectrum strictly in the left half-plane. Jordan decomposition supplies an exponential bound, with \(g\) chosen smaller than the spectral decay rate to absorb possible polynomial factors. Apply this bound to \(S_0-\Id_N/N\). The limiting matrix has no off-diagonal entries; the entanglement formulas vanish and the displayed mixture is separable. \end{proof} \begin{remark}[The localized hypothesis matters] A general mixed \(S_0\) need not generate entanglement: \(S_0=\Id_N/N\) is already stationary conditionally. The generation theorem is therefore not asserted for all one-excitation density matrices. With no dissipation, decay also need not occur; the disorder-free two-site chain has \(C_{01}(\tau)=|\sin(2\tau)|\). \end{remark} \begin{theorem}[Finite-window unitary limit] \label{thm:largechi} Fix \(N,S_0,\widetilde h,r\), and let \[ S_{\mathrm u}(\tau)=e^{-\ii\tau\widetilde h}S_0 e^{\ii\tau\widetilde h}, \qquad \rho_{\mathrm u}=VS_{\mathrm u}V^\dagger. \] For finite \(\chi>0\), \begin{align} \norm{S(\tau)-S_{\mathrm u}(\tau)}_1 &\le\frac{4(1-r)\tau}{\chi},\label{eq:Sunit}\\ \trdist(\rho(\tau),\rho_{\mathrm u}(\tau)) &\le\min\{1,2\tau/\chi\}.\label{eq:unitbound} \end{align} All diagnostics above converge uniformly to their unitary counterparts on every fixed \([0,T]\). In particular, \[ \sup_{0\le\tau\le T}|C_{ij}-C_{ij}^{\mathrm u}| \le\min\{1,4T/\chi\}. \] \end{theorem} \begin{proof} Variation of constants with the unitary propagator \(\mathcal U\) gives \[ S(\tau)-S_{\mathrm u}(\tau) =-\kappa\int_0^\tau \mathcal U_{\tau-s}(S(s)-\Delta S(s))\dd s. \] Both matrices in the integrand difference are density matrices, so its trace norm is at most two. This proves \eqref{eq:Sunit}. Reconstruction gives \[ \trdist(\rho,\rho_{\mathrm u}) \le (1-q)+q\,\trdist(S,S_{\mathrm u}) \le\frac{2r\tau}{\chi}+\frac{2(1-r)\tau}{\chi}. \] For Hermitian \(M\) and distinct orthonormal \(u,v\), \(2|\bra uM\ket v|\le\norm M_1\), by testing against a phase-aligned Hermitian off-diagonal operator of norm one. Together with \eqref{eq:C}, this proves the concurrence bound. The remaining diagnostics are continuous functions of \(q,S\); uniform continuity on their compact finite-dimensional state domain gives the stated uniform convergence. \end{proof} \begin{proposition}[Deterministic detuning stability] \label{prop:disorder} For two models with the same \(J,\gr,\gd,S_0\), and Hamiltonians \(\widetilde h,\widetilde h'\), \[ \trdist(\rho(\tau),\rho'(\tau)) \le q(\tau)\min\left\{1,\, \tau\inf_{c\in\R} \norm{\widetilde h-\widetilde h'-c\Id}_\infty\right\}. \] \end{proposition} \begin{proof} Duhamel's formula has Hermitian source \(-\ii[\widetilde h-\widetilde h',S'(s)]\), of trace norm at most \(2\norm{\widetilde h-\widetilde h'-c\Id}_\infty\). Contractivity from Lemma~\ref{lem:physical} proves the conditional distance bound. The vacuum weights agree, so reconstruction multiplies the conditional distance by \(q\). \end{proof} \begin{remark}[Limits of these comparisons] The finite-window estimate is not a fitted saturation law, global monotonicity theorem, or uniform large-time limit. The detuning bound does not establish a sharp disorder threshold. Ensemble crossover statements require a distribution and statistic. Protocols A and B answer different observation questions. \end{remark} \begin{proposition}[Two-site closed-form benchmark] \label{prop:twosite} For the localized preparation, \(N=2\), and equal detunings, let \(\kappa=2(1-r)/\chi\) and define \[ F_\kappa(\tau)= \begin{cases} \sin(\omega\tau)/\omega, &\kappa<4,\quad \omega=\sqrt{4-\kappa^2/4},\\ \tau,&\kappa=4,\\ \sinh(\nu\tau)/\nu, &\kappa>4,\quad \nu=\sqrt{\kappa^2/4-4}. \end{cases} \] Then \[ C_{01}(\tau)=2e^{-2r\tau/\chi}e^{-\kappa\tau/2} |F_\kappa(\tau)|. \] For \(r=1\), this is \(e^{-2\tau/\chi}|\sin(2\tau)|\). \end{proposition} \begin{proof} Write \(S_{01}=\ii y\), \(z=S_{00}-S_{11}\). The equations are \[ z'=-4y,\qquad y'=z-\kappa y,\qquad z(0)=1,\ y(0)=0. \] Hence \(y''+\kappa y'+4y=0\), \(y'(0)=1\), whose solution is \(y=e^{-\kappa\tau/2}F_\kappa(\tau)\). Use \(C=2q|y|\). \end{proof} For stable overdamped evaluation one can instead use the identical expression \[ e^{-\kappa\tau/2}F_\kappa(\tau) =\frac{e^{-(\kappa/2-\nu)\tau} -e^{-(\kappa/2+\nu)\tau}}{2\nu}. \] This is an analytical identity, not an executed numerical result. \section{Finite local instruments and typed records} \label{sec:instruments} The ordinary finite-channel framework is described in \cite{Watrous}. The identities needed here are proved explicitly. \begin{definition}[Specified instrument] A finite instrument from \(\Hil_{\mathrm{in}}\) to \(\Hil_{\mathrm{out}}\) has Kraus operators \(M_{a\mu}\) with \[ E_a=\sum_\mu M_{a\mu}^\dagger M_{a\mu},\qquad \sum_aE_a=\Id_{\mathrm{in}}. \] Set \[ \Phi_a(M)=\sum_\mu M_{a\mu}MM_{a\mu}^\dagger,\qquad \widehat\Phi(M)=\sum_a\kb aa_R\otimes\Phi_a(M). \] The record includes every outcome, including failure labels. \end{definition} \begin{theorem}[Instrument consistency and no-signaling] \label{thm:instruments} For a density matrix \(\rho\): \begin{enumerate}[label=\textup{(\roman*)}] \item \(p_a=\Tr\Phi_a(\rho)=\Tr(E_a\rho)\ge0\), and \(\sum_ap_a=1\). \item If \(p_a>0\), the conditional state is \(\Phi_a(\rho)/p_a\). If \(p_a=0\), then \(\Phi_a(\rho)=0\); no normalized branch is assigned. \item \(\widehat\Phi\) is CPTP. \item Actual tensor-local instruments on disjoint factors commute as selective superoperators after identifying output types. \item A remote trace-preserving channel leaves local outcome probabilities unchanged, including for entangled inputs. \end{enumerate} \end{theorem} \begin{proof} Kraus positivity, cyclicity, and completeness prove (i). A positive matrix of trace zero is zero, proving (ii). The record Kraus operators \(\ket a_R\otimes M_{a\mu}\) are complete, proving (iii). For separate factors, composition in either order gives the same Kraus operator \(M_{a\mu}\otimes K_{b\nu}\), proving (iv). For a remote channel with \(\sum_bK_b^\dagger K_b=\Id_B\), \[ \begin{aligned} p'_a &=\sum_b\Tr\!\left[ (E_a\otimes\Id)(\Id\otimes K_b) \rho(\Id\otimes K_b^\dagger)\right]\\ &=\Tr\!\left[ \left(E_a\otimes\sum_bK_b^\dagger K_b\right)\rho\right] =\Tr[(E_a\otimes\Id)\rho]. \end{aligned} \] \end{proof} \begin{remark}[Required implementation checks] For a supplied single update operator, require \(M_a^\dagger M_a=E_a\); for several operators use their sum. Declared footprints do not establish tensor locality. Remote postselection is not a trace-preserving unconditioned channel. Finally, equality of effects does not imply equality of instruments. \end{remark} \begin{theorem}[Typed deferred readout] \label{thm:deferral} For a measured factor \(S\), complement \(B\), and computational basis \(\ket z_S\), define \[ V_{\mathrm{rec}}(\ket z_S\ket b_B) =\ket z_S\ket z_A\ket b_B. \] Reading \(A\) computationally and retaining \(S\) gives \[ \mathcal I^{\mathrm{nd}}(M) =\sum_z\kb zz_R\otimes P_zMP_z,\qquad P_z=\kb zz_S\otimes\Id_B. \] Reading \(A\) and discarding \(S\) gives instead \[ \mathcal M(M) =\sum_z\kb zz_R\otimes (\bra z_S\otimes\Id_B)M(\ket z_S\otimes\Id_B). \] These identities hold with any external reference and with the same fixed continuation channel on \(B\) in both experiments. \end{theorem} \begin{proof} Expand \(M=\sum_{z,w}\kb zw_S\otimes M_{zw}\). Coherent recording gives \(\sum_{z,w}\kb zw_S\otimes\kb zw_A\otimes M_{zw}\). Reading \(A\) removes the \(z\ne w\) terms and produces \(\sum_z\kb zz_R\otimes\kb zz_S\otimes M_{zz}\). Tracing out \(S\) gives the destructive formula. A reference simply enlarges the untouched matrix coefficients. A fixed channel on \(B\) replaces each \(M_{zw}\) by its image. \end{proof} Coherent recording itself is not a classical-record channel. The output types above differ. Arbitrary later interactions with the retained measured factor, carrier, or outcome-dependent feedback require additional hypotheses. \section{Compressed oscillator position measurements} \label{sec:bosonic} \subsection{Exact objects and state approximation} Let \(\Hil_{\mathrm{osc}}=L^2(\R)\) have oscillator number basis \(\ket n\), \(n\ge0\). In this section \(a\) denotes the oscillator annihilation operator, not the two-level matrix of Definition~\ref{def:model}. Let \[ X_{\mathrm{osc}}=\frac{a+a^\dagger}{\sqrt2} \] be multiplication by \(x\), with PVM \[ (E(B)\psi)(x)=\mathbf1_B(x)\psi(x). \] For an integer \(d\ge1\), define \[ P_d=\sum_{n=0}^{d-1}\kb nn,\quad \Hil_d=P_d\Hil_{\mathrm{osc}},\quad E_d(B)=P_dE(B)P_d|_{\Hil_d}. \] \begin{theorem}[Compressed-position POVM] \label{thm:compression} The compressed effects form a countably additive normalized POVM: \[ E_d(B)\ge0,\qquad E_d(\R)=\Id_d. \] Their sum over a countable measurable partition is \(\Id_d\). They are generally not projections and are not generally the spectral effects of \(P_dX_{\mathrm{osc}}P_d\). \end{theorem} \begin{proof} Positivity follows by testing on vectors in \(\Hil_d\). Compression preserves countable additivity of matrix elements, and \(P_dE(\R)P_d=P_d\). For \(d=1\), the vacuum position density is \(\pi^{-1/2}e^{-x^2}\), giving \(E_1((-\infty,0])=\Id_1/2\), not a projection. The compressed position operator is zero, whose spectral effect for this set is instead \(\Id_1\). \end{proof} \begin{lemma}[Normalized projection error] \label{lem:projection} For any oscillator state \(\rho\), suppose \[ \delta_d=\Tr((\Id-P_d)\rho)<1,\qquad \rho_d=\frac{P_d\rho P_d}{1-\delta_d}. \] Embedding \(\rho_d\) into the full space, \[ \trdist(\rho,\rho_d)\le\sqrt{\delta_d}. \] For any finite state \(\sigma_d\), \begin{equation} \trdist(\rho,\sigma_d) \le\sqrt{\delta_d}+\trdist(\rho_d,\sigma_d). \label{eq:stateerror} \end{equation} Every effect-probability difference is bounded by the corresponding trace distance. \end{lemma} \begin{proof} A purification \(\ket\Psi\) of \(\rho\) and its normalized projection have overlap \(\sqrt{1-\delta_d}\). Their pure-state trace distance is \(\sqrt{\delta_d}\), by diagonalizing their difference on their two-dimensional span. Partial trace is contractive, proving the first assertion. The triangle inequality proves \eqref{eq:stateerror}. For an effect \(0\le F\le\Id\), the Jordan decomposition of a trace-zero state difference gives \(|\Tr(F(\rho-\sigma))|\le\trdist(\rho,\sigma)\). \end{proof} Normalized projection is a preparation comparison, not a claim that postselected normalization defines a linear deterministic channel. \begin{proposition}[Coherent-state truncation and displacement] \label{prop:coherent} For finite \(\alpha\in\C\), let \(\mu=|\alpha|^2\) and \[ \ket\alpha=e^{-\mu/2} \sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n. \] Its omitted probability is \[ \delta_d=1-e^{-\mu}\sum_{n=0}^{d-1}\frac{\mu^n}{n!}. \] For \(\mu>0\) and \(d>\mu\), \[ \delta_d\le e^{-\mu}(e\mu/d)^d; \] for \(\mu=0\), \(\delta_d=0\). Define \[ a_d=P_daP_d|_{\Hil_d},\quad \ket{\beta_d}=e^{\alpha a_d^\dagger-\overline\alpha a_d}\ket0, \quad \ket{\alpha_d}=\frac{P_d\ket\alpha}{\sqrt{1-\delta_d}}. \] These preparations need not agree, but \begin{equation} \trdist(\kb{\beta_d}{\beta_d},\kb{\alpha_d}{\alpha_d}) \le\min\{1,I_d(\alpha)+1-\sqrt{1-\delta_d}\}, \label{eq:dispbound} \end{equation} where \[ I_d(\alpha)=|\alpha|\sqrt d\int_0^1 e^{-s^2\mu/2}\frac{(s|\alpha|)^d}{\sqrt{d!}}\dd s. \] \end{proposition} \begin{proof} The squared coefficients are Poisson probabilities. For a Poisson variable \(K\), \(\Pr(K\ge d)\le\exp(\mu(e^s-1)-sd)\). Choose \(s=\log(d/\mu)>0\). For displacement, write \(A=\alpha a^\dagger-\overline\alpha a\), \(A_d=P_dAP_d|_{\Hil_d}\), and \(u(s)=P_d\ket{s\alpha}\). Differentiation gives \[ u'=A_du+P_dA(\Id-P_d)\ket{s\alpha}, \] with boundary source \[ -\overline\alpha\sqrt d\, e^{-s^2\mu/2}\frac{(s\alpha)^d}{\sqrt{d!}}\ket{d-1}. \] Variation of constants and unitarity of \(e^{sA_d}\) give \(\norm{\ket{\beta_d}-P_d\ket\alpha}\le I_d(\alpha)\). Normalization adds vector distance \(1-\sqrt{1-\delta_d}\). Pure-state trace distance is at most the vector distance, proving \eqref{eq:dispbound}. \end{proof} \subsection{Composite trapezoidal approximation} Let \[ \phi_n(x)= \frac{H_n(x)e^{-x^2/2}}{\pi^{1/4}\sqrt{2^n n!}},\qquad H_n(x)=(-1)^ne^{x^2}\frac{\mathrm d^n}{\mathrm dx^n}e^{-x^2}. \] Set \[ v_d(x)=(\phi_0(x),\ldots,\phi_{d-1}(x))^{\mathsf T}, \qquad G_d(x)=v_d(x)v_d(x)^\dagger. \] For real \(x\), \(G_d(x)\ge0\), and \(E_d(B)=\int_BG_d(x)\dd x\). Fix \(L>0\). Suppose \(B\cap[-L,L]\) is a finite union of intervals, partitioned into cells \(I_\ell=[a_\ell,b_\ell]\) of widths \(h_\ell=b_\ell-a_\ell\). Boundary points have zero position probability and may be assigned consistently to either bin. Define the exact-arithmetic trapezoidal matrix \begin{equation} Q_d^{\mathrm{trap}}(B;L) =\sum_\ell\frac{h_\ell}{2} (G_d(a_\ell)+G_d(b_\ell)), \label{eq:trapezoid} \end{equation} and \begin{equation} \eta_d^{\mathrm{trap}}(B;L) =\frac1{12}\sum_\ell h_\ell^3 \sup_{x\in I_\ell}\norm{G_d''(x)}_\infty. \label{eq:trapbudget} \end{equation} This rule retains positive weights and half-weight endpoints. It is the quadrature to be checked against the reported simulator. \begin{theorem}[Trapezoidal operator-error bound] \label{thm:trap} The matrix in \eqref{eq:trapezoid} is positive and \begin{equation} \norm{Q_d^{\mathrm{trap}}(B;L) -E_d(B\cap[-L,L])}_\infty \le\eta_d^{\mathrm{trap}}(B;L). \label{eq:traperror} \end{equation} \end{theorem} \begin{proof} Positivity follows from positive weights. Twice integrating by parts, componentwise, gives for one cell \[ \int_a^bG_d(x)\dd x-\frac h2(G_d(a)+G_d(b)) =-\frac12\int_a^b(x-a)(b-x)G_d''(x)\dd x. \] Since \(\frac12\int_a^b(x-a)(b-x)\dd x=h^3/12\), taking operator norms and summing proves the claim. \end{proof} \begin{remark}[Midpoint alternative] The earlier midpoint construction is also valid as a different rule: \[ Q_d^{\mathrm{mid}}=\sum_\ell h_\ell G_d((a_\ell+b_\ell)/2), \qquad \eta_d^{\mathrm{mid}} =\frac1{24}\sum_\ell h_\ell^3\sup_{I_\ell}\norm{G_d''}_\infty. \] Taylor expansion at each midpoint proves this bound. A midpoint certificate must not be attached to a trapezoidal implementation. \end{remark} \begin{lemma}[A cellwise derivative-enclosure interface] \label{lem:Gbound} On a cell \(I\), suppose validated bounds satisfy \[ \sup_I|\phi_n^{(j)}|\le U_{n,j}, \qquad c_j=\left(\sum_{n=0}^{d-1}U_{n,j}^2\right)^{1/2}, \qquad j=0,1,2. \] Then \[ \sup_I\norm{G_d''}_\infty\le2c_0c_2+2c_1^2. \] \end{lemma} \begin{proof} Differentiate to obtain \[ G_d''=v_d''v_d^\dagger+2v_d'(v_d')^\dagger +v_d(v_d'')^\dagger. \] A rank-one matrix \(uv^\dagger\) has operator norm \(\norm u\,\norm v\). The triangle inequality proves the bound. \end{proof} The \(U_{n,j}\) must enclose entire cells, not just sampled values. They can be obtained from outward-rounded evaluations of the Hermite formula and its derivatives, with subdivision when necessary. The computable bounds in Lemma~\ref{lem:Gbound} remove the need for an unverified sampled operator-norm supremum. Arbitrary-precision ball-arithmetic integration with rigorous error bounds is an established methodology \cite{Johansson}. That reference supplies a general numerical method, not an executed certificate for this manuscript. If complex-analytic integration is used for oscillator entries, the holomorphic integrand is \(\phi_m(z)\phi_n(z)\), whose restriction to the real axis is the required product. Extending a real-axis conjugate expression as \(v(z)v(z)^\dagger\) would not give a holomorphic integrand. \subsection{Separate preparation, tail, quadrature, and arithmetic errors} \begin{theorem}[Probability comparison with arithmetic error] \label{thm:bosonerror} Let \(\rho\) be an oscillator state, \(\sigma_d\) a density matrix on \(\Hil_d\), and \(A_d(B;L)\) a Hermitian numerical matrix with a certified bound \[ \norm{A_d(B;L)-Q_d^{\mathrm{trap}}(B;L)}_\infty \le\alpha_d(B;L). \] Then \begin{align} &|\Tr(\rho E(B))-\Tr(\sigma_dA_d(B;L))| \nonumber\\ &\quad\le \underbrace{\trdist(\rho,\sigma_d)}_{\text{preparation}} +\underbrace{ \Tr(\sigma_dE_d(\R\setminus[-L,L])) }_{\text{domain tail}} +\underbrace{\eta_d^{\mathrm{trap}}(B;L)}_{\text{quadrature}} +\underbrace{\alpha_d(B;L)}_{\text{arithmetic}}. \label{eq:bosonbudget} \end{align} The tail term is bounded by \begin{equation} \min\left\{1,\frac{\Tr(\sigma_dX_{\mathrm{osc}}^2)}{L^2}\right\} \le\min\left\{1,\frac{2d-1}{L^2}\right\}. \label{eq:tail} \end{equation} The preparation term can use \eqref{eq:stateerror} and \eqref{eq:dispbound}. A separately rounded scalar evaluation of the trace requires its own additional certified error. \end{theorem} \begin{proof} Insert the exact probabilities for \(\sigma_d\) on \(B\), then on \(B\cap[-L,L]\), followed by the exact trapezoidal value. The effect inequality in Lemma~\ref{lem:projection}, the outside-domain probability, Theorem~\ref{thm:trap}, and the arithmetic norm bound control the four differences. The pointwise inequality \(\mathbf1_{\{|x|>L\}}\le x^2/L^2\) proves the first tail bound. For a vector \(\psi\in\Hil_d\), \[ \norm{X_{\mathrm{osc}}\psi}^2 \le\norm{a\psi}^2+\norm{a^\dagger\psi}^2 =\bra\psi(2a^\dagger a+\Id)\ket\psi \le(2d-1)\norm\psi^2. \] A convex decomposition proves the mixed-state bound. \end{proof} \begin{remark}[Raw numerical values need not be probabilities] Exact trapezoidal matrices are positive, but may exceed the identity or fail bin completeness. Rounded matrices may even lose positivity. Thus the raw scalar in \eqref{eq:bosonbudget} is an approximation, not automatically a measurement probability. No numerical clipping is implicit in the theorem. \end{remark} \subsection{Arithmetic-aware completion to a genuine POVM} \begin{theorem}[Completed POVM with a bin-independent error bound] \label{thm:completion} Partition \([-L,L]\) into \(m\) interval bins \(B_k\), and let \[ E_k=E_d(B_k),\quad Q_k=Q_d^{\mathrm{trap}}(B_k;L). \] Suppose \(A_k=A_k^\dagger\) and certified nonnegative bounds satisfy \[ \norm{A_k-Q_k}_\infty\le\alpha_k,\qquad \norm{Q_k-E_k}_\infty\le\eta_k. \] Define \begin{equation} P_k=A_k+\alpha_k\Id_d,\qquad \varepsilon_k=\eta_k+2\alpha_k,\qquad \eta=\sum_k\varepsilon_k, \label{eq:arithrepair} \end{equation} and \begin{equation} F_k=\frac{P_k}{1+\eta},\qquad F_{\mathrm{out}}=\Id_d-\sum_kF_k. \label{eq:completion} \end{equation} These are the effects of a valid POVM. For each interior bin, \[ \norm{F_k-E_k}_\infty \le\frac{\varepsilon_k+\eta}{1+\eta}. \] For every finite input state, the total variation distance from the exact compressed-position law, including the outside bin, is at most \begin{equation} \boxed{\tv\le\min\left\{1,\frac{2\eta}{1+\eta}\right\}.} \label{eq:TVcompletion} \end{equation} For a full-space state \(\rho\) and finite preparation \(\sigma_d\), comparison with the full exact position law for the same bins adds at most \(\trdist(\rho,\sigma_d)\). \end{theorem} \begin{proof} The arithmetic bound gives \(A_k\ge Q_k-\alpha_k\Id_d\), so \(P_k\ge Q_k\ge0\). Also \(\norm{P_k-E_k}_\infty\le\varepsilon_k\). Writing \(P=\sum_kP_k\) and \(E=\sum_kE_k\le\Id_d\), \[ P\le E+\eta\Id_d\le(1+\eta)\Id_d. \] Thus \(F_{\mathrm{out}}\ge0\), and the effects sum to the identity. The individual norm estimate follows from \[ F_k-E_k=\frac{P_k-E_k-\eta E_k}{1+\eta}. \] For a finite input state, put \(p_k=\Tr(\sigma_dE_k)\) and \(u_k=\Tr(\sigma_dP_k)\). Then \[ |u_k-p_k|\le\varepsilon_k,\qquad \sum_kp_k\le1. \] The completed interior probability difference is \[ d_k=\frac{u_k-p_k-\eta p_k}{1+\eta}. \] Consequently \[ \sum_k|d_k| \le\frac{\sum_k\varepsilon_k+\eta\sum_kp_k}{1+\eta} \le\frac{2\eta}{1+\eta}. \] The outside difference is \(-\sum_kd_k\), so total variation is no greater than \(\sum_k|d_k|\). It is also at most one. Finally, applying any exact POVM contracts state trace distance to outcome total variation; the triangle inequality proves the last assertion. \end{proof} \begin{remark}[What is and is not completed] The shift in \eqref{eq:arithrepair} is essential unless positivity of the numerical \(A_k\) has been independently certified. Scaling an arbitrary rounded matrix alone cannot repair a negative eigenvalue. The displayed effects are exact mathematical matrices defined by the stored centers and certified bounds. Exact rational centers and rational outward bounds permit an exact rational completion. If the completed effects or reported probabilities are subsequently rounded, those errors require additional enclosures. One possible finite instrument uses update operators \(\sqrt{F_k}\). The effect-law estimate above does not, by itself, certify an approximation of a particular post-measurement instrument. Because the exact outside effect is included, its probability is retained rather than postselected away. The domain-tail bound is needed when outside events are omitted or when a truncated raw quadrature is compared as in \eqref{eq:bosonbudget}. \end{remark} \section{Validated dynamical residual interfaces} \label{sec:residual} \subsection{Exact vectorized generator} With column-stacking, \(\vecop(AMB)=(B^{\mathsf T}\otimes A)\vecop M\). If \(\Pi_\Delta\) projects onto diagonal entries, then \[ \frac{\mathrm d}{\mathrm dt}\vecop R = \left[ -\ii(\Id_N\otimes h-h^{\mathsf T}\otimes\Id_N) -\gr\Id_{N^2} -2\gd(\Id_{N^2}-\Pi_\Delta) \right]\vecop R. \] The matrix-exponential representation is exact but not an execution claim. Storing \(S\) requires \(N^2\) entries; a dense \(N^2\times N^2\) generator need not be stored. \subsection{Accuracy without a positivity assumption} \begin{theorem}[Cellwise a posteriori residual bound] \label{thm:residual} Fix the scaled generator \[ \mathcal A(M)=-\ii[\widetilde h,M] -\kappa(M-\Delta M),\qquad \kappa=2(1-r)/\chi. \] Let \(0=t_0