Unified Measurement via Windowed Readout:\\ Born Probability = Minimal KL,\\ Pointer Basis = Minimal Energy Eigenbasis\\[10pt] \large (With Non-Asymptotic Error Closure and Window/Kernel Optimization)
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2025
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| _version_ | 1866902028114984960 |
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| author | Ma, Haobo Zhang, Wenlin |
| author_facet | Ma, Haobo Zhang, Wenlin |
| contents | Within unified framework of mirror kernel--de Branges--Kreĭn canonical system--information geometry, this paper proposes and rigorously proves three main theorems: 1. Windowed Readout Theorem: Any realizable quantum measurement readout equivalent to weighting of (relative or absolute) local density of states (LDOS) by ``energy window w_R and frontend kernel h''; when adopting realistic discrete sampling--finite truncation procedure, error can be non-asymptotically closed by Nyquist (alias)--Poisson (sampling)--Euler--Maclaurin (EM, sum--integral difference) three terms, with alias term strictly zero under bandlimited + Nyquist conditions. Conclusion based on Herglotz property and boundary value dictionary (\Im m(E+i0)=\pi\rho(E)) of Weyl--Titchmarsh m-function and its equivalent formulation with canonical systems. 2. Born Probability = Minimal KL (Information Projection): When readout dictionary aligns with log-partition potential \Lambda(\rho)=\log\!\sum_j w_j e^{\langle\beta_j,\rho\rangle}, minimal energy projection with unit response equivalent to minimal Kullback--Leibler (KL) divergence under linear moment constraints; softmax probability precisely minimal-KL projection weights, converging via \Gamma-limit to hard projection (Hilbert orthogonal) as softening parameter \tau\!\downarrow\!0 (equivalently inverse temperature \kappa=1/\tau\!\uparrow\!\infty). Equivalently using Fenchel--Legendre duality / Bregman--KL identity / Csiszár I-projection. 3. Pointer Basis = Eigenbasis of Minimal Energy/Information Projection: Under finite dictionary, coefficient vector of minimal energy mollifier \beta^\star=G^{-1c}{c^\ast G^{-1}c}; in Gram spectral decomposition G=U\Lambda U^\ast, \beta^\star expanded along \{u_k\} weighted by \lambda_k^{-1}, thus direction contributing strongest to \beta^\star realized by $ \arg\max_k\ |\langle u_k,c\rangle|^2{\lambda_k}; small eigenvalue trend amplifies that direction, but whether dominates depends on simultaneously having sufficiently large projection |\langle u_k,c\rangle|. Soft version information Hessian \nabla^2\Lambda spectral basis isomorphic to this. On scattering side, via Birman--Kreĭn and Wigner--Smith standard construction, single-channel phase derivative and (relative) spectral density satisfy \varphi'(E)=\pi\,\rho_{rel}(E)=\pi\,\xi'(E),\qquad Q(E)=-i\,S(E)^\daggerdS{dE}, hence 1{2\pi}trQ(E)=\rho_{rel}(E)$. This interprets ``negative delay'' as result of reference choice and relative counting, not causality violation. Keywords: Windowed readout; Weyl--Titchmarsh; spectral shift function; Wigner--Smith time delay; de Branges space; BN--Bregman; minimal KL; PSWF; Nyquist--Poisson--EM; non-asymptotic error. |
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| id | zenodo_https___doi_org_10_5281_zenodo_17697593 |
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| publishDate | 2025 |
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| spellingShingle | Unified Measurement via Windowed Readout:\\ Born Probability = Minimal KL,\\ Pointer Basis = Minimal Energy Eigenbasis\\[10pt] \large (With Non-Asymptotic Error Closure and Window/Kernel Optimization) Ma, Haobo Zhang, Wenlin Unified Time Scale Generalized Entropy Quantum Scattering General Relativity Boundary Time Geometry Causal Structure Information Theory Wigner-Smith Time Delay Modular Flow QNEC Spectral Shift Function Time Geometry Within unified framework of mirror kernel--de Branges--Kreĭn canonical system--information geometry, this paper proposes and rigorously proves three main theorems: 1. Windowed Readout Theorem: Any realizable quantum measurement readout equivalent to weighting of (relative or absolute) local density of states (LDOS) by ``energy window w_R and frontend kernel h''; when adopting realistic discrete sampling--finite truncation procedure, error can be non-asymptotically closed by Nyquist (alias)--Poisson (sampling)--Euler--Maclaurin (EM, sum--integral difference) three terms, with alias term strictly zero under bandlimited + Nyquist conditions. Conclusion based on Herglotz property and boundary value dictionary (\Im m(E+i0)=\pi\rho(E)) of Weyl--Titchmarsh m-function and its equivalent formulation with canonical systems. 2. Born Probability = Minimal KL (Information Projection): When readout dictionary aligns with log-partition potential \Lambda(\rho)=\log\!\sum_j w_j e^{\langle\beta_j,\rho\rangle}, minimal energy projection with unit response equivalent to minimal Kullback--Leibler (KL) divergence under linear moment constraints; softmax probability precisely minimal-KL projection weights, converging via \Gamma-limit to hard projection (Hilbert orthogonal) as softening parameter \tau\!\downarrow\!0 (equivalently inverse temperature \kappa=1/\tau\!\uparrow\!\infty). Equivalently using Fenchel--Legendre duality / Bregman--KL identity / Csiszár I-projection. 3. Pointer Basis = Eigenbasis of Minimal Energy/Information Projection: Under finite dictionary, coefficient vector of minimal energy mollifier \beta^\star=G^{-1c}{c^\ast G^{-1}c}; in Gram spectral decomposition G=U\Lambda U^\ast, \beta^\star expanded along \{u_k\} weighted by \lambda_k^{-1}, thus direction contributing strongest to \beta^\star realized by $ \arg\max_k\ |\langle u_k,c\rangle|^2{\lambda_k}; small eigenvalue trend amplifies that direction, but whether dominates depends on simultaneously having sufficiently large projection |\langle u_k,c\rangle|. Soft version information Hessian \nabla^2\Lambda spectral basis isomorphic to this. On scattering side, via Birman--Kreĭn and Wigner--Smith standard construction, single-channel phase derivative and (relative) spectral density satisfy \varphi'(E)=\pi\,\rho_{rel}(E)=\pi\,\xi'(E),\qquad Q(E)=-i\,S(E)^\daggerdS{dE}, hence 1{2\pi}trQ(E)=\rho_{rel}(E)$. This interprets ``negative delay'' as result of reference choice and relative counting, not causality violation. Keywords: Windowed readout; Weyl--Titchmarsh; spectral shift function; Wigner--Smith time delay; de Branges space; BN--Bregman; minimal KL; PSWF; Nyquist--Poisson--EM; non-asymptotic error. |
| title | Unified Measurement via Windowed Readout:\\ Born Probability = Minimal KL,\\ Pointer Basis = Minimal Energy Eigenbasis\\[10pt] \large (With Non-Asymptotic Error Closure and Window/Kernel Optimization) |
| topic | Unified Time Scale Generalized Entropy Quantum Scattering General Relativity Boundary Time Geometry Causal Structure Information Theory Wigner-Smith Time Delay Modular Flow QNEC Spectral Shift Function Time Geometry |
| url | https://doi.org/10.5281/zenodo.17697593 |