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arXiv:2512.15166

Composite N-Q-S: Serial/Parallel Instrument Axioms, Bipartite Order-Effect Bounds, and a Monitored Lindblad Limit

Kazuyuki Yoshida

quant-phmath-phSubmitted Dec 17, 2025• Announced Dec 18, 2025• 0 citations
arXivHTMLPDF

Abstract

We develop a composite operational architecture for sequential quantum measurements that (i) gives a tight bipartite order-effect bound with an explicit equality set characterized on the Halmos two-subspace block, (ii) upgrades Doeblin-type minorization to composite instruments and proves a product lower bound for the operational Doeblin constants, yielding data-driven exponential mixing rates, (iii) derives a diamond-norm commutator bound that quantifies how serial and parallel rearrangements influence observable deviations, and (iv) establishes a monitored Lindblad limit that links discrete look-return loops to continuous-time GKLS dynamics under transparent assumptions. Building on the GKLS framework of Gorini, Kossakowski, Sudarshan, Lindblad, Davies, Spohn, and later work of Fagnola-Rebolledo and Lami et al., we go beyond asymptotic statements by providing finite-sample certificates for the minorization parameter via exact binomial intervals and propagating them to rigorous bounds on the number of interaction steps required to attain a prescribed accuracy. A minimal qubit toy model and CSV-based scripts are supplied for full reproducibility. Our results position order-effect control and operational mixing on a single quantitative axis, from equality windows for pairs of projections to certified network mixing under monitoring. The framework targets readers in quantum information and quantum foundations who need explicit constants that are estimable from data and transferable to device-level guarantees.

Comments
12 pages, 1 figure, 1 table. Data and code available at Zenodo, DOI: 10.5281/zenodo.17959208

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We study sequential quantum measurements within an operational N–Q–S architecture and prove three main results. (i) A tight order-effect bound with saturation conditions characterized by Halmos two-subspace blocks; the construction extends to POVMs via Naimark dilation. (ii) A Doeblin-type minorization for instruments, yielding a quantitative 1-norm contraction and an exponential mixing rate γ ≥ −ln(1 − δ)/Δt for the look–return loop. (iii) A monitored discrete-to-GKLS bridge: as Δt → 0 under standard stability assumptions, the certified rate transfers to a compatible GKLS semigroup. We define testable indices (ONS/PII/PAD) linking our bounds to behavioral order effects, and we supply complete proofs (main text/appendices) together with a reproducibility package (code, data, and figure scripts).

•Oct 23, 2025•0 citations

Data-certified mixing-rate bounds and GKLS identification from one-step channels

Yoshida, Kazuyuki

Many tasks in open quantum systems and quantum information theory require guarantees on the mixing rate of a continuous-time Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) semigroup, while experimental access is typically limited to a discrete one-step completely positive trace-preserving (CPTP) map estimated from data. We provide an operational route from such one-step information to certified lower bounds on exponential mixing rates and to a compatible Lindblad generator. Given a statistically consistent estimate of a stationary channel E_delta t and one stationary state tau, we first certify a Doeblin minorization mass for E_delta t and translate it into a data-certified contraction rate gamma_op in trace distance. A Chernoff-type telescoping argument then controls the embedding error between discrete dynamics and any continuous-time realization, showing that the same rate holds up to an O(t Delta t) correction. Finally, we recover a GKLS generator by a convex semidefinite log-projection onto the generator cone, constrained to be consistent with the observed invariant state and the certificate. The resulting guarantee is non-asymptotic, depends only on observed counts and one stationary state, and bridges classical Doeblin theory with data-driven identification of quantum Markov semigroups from one-step CPTP experiments.

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Quantum Measurement Without an External Observer in OT-GKSL Certified Reference Frames, Finite Entropic Readout, and Noether Balance Laws

Gwenolé Bocquet

Pedagogical Guide, Foundations Audit, Uniqueness & Substitutability Analysis, Literature Comparison, and Reading Guidance : A Pedagogical Guide to Understanding the OT–GKSL Architecture This paper develops a relational axiomatics of quantum measurement within the Einstein-locked OT/GKSL source-readout architecture. Its central question is: under what physical conditions can one quantum system function as the measurement frame of another? The native dynamics is described on finite effective quantum supports by density operators and completely positive Markovian evolution of GKSL type (Gorini, Kossakowski & Sudarshan, 1976; Lindblad, 1976). At this level, the labels assigned to two subsystems do not by themselves determine which is the apparatus, the observer, or the measured object. This starting point is consonant with operational quantum measurement theory, in which measurements are represented by physical instruments, effects, and state transformations rather than by an unanalyzed external observer (Davies & Lewis, 1970; Ozawa, 1984), and with quantum-reference-frame approaches in which reference systems may themselves be finite quantum systems (Bartlett, Rudolph & Spekkens, 2007; Giacomini, Castro-Ruiz & Brukner, 2019). The specifically OT–GKSL proposal is that a measurement frame is a certified record-bearing subsystem. Given two finite systems A and B, the directed statement A←B means that state-dependent content associated with B is stably and physically encoded in records carried by A. The direction is not inferred from subsystem labels, entropy ordering, or correlation alone. It requires a declared record algebra or instrument, operationally accessible record channels, stability under the relevant open-system dynamics, sufficient distinguishability or identifiability of the encoded alternatives, and controlled native-to-readout bridge diagnostics. A measurement outcome label is not yet a physical record. For a quantum instrument {Iy}, a physical readout is represented by an explicit record register, for example MR(ρ)=∑yIy(ρ)⊗∣y⟩⟨y∣R, after which accessibility, persistence, distinguishability, and stability of the register can be tested. This distinction follows the operational treatment of quantum instruments (Davies & Lewis, 1970; Ozawa, 1984) while making the physical record requirement explicit inside the OT–GKSL architecture. The paper also clarifies the role of entropy in measurement. Neither S(A), S(B), nor the raw difference S(B)−S(A) determines the measurement direction. Mutual information I(A:B)=S(A)+S(B)−S(AB) diagnoses total correlation but is symmetric and therefore does not select a readout orientation. Quantum conditional entropy S(B∣A)=S(AB)−S(A) is relational but remains distinct from the classical conditional entropy obtained after a physical outcome register has formed, S(B∣Y)=∑ypyS(ρB∣y). These distinctions are standard in quantum information theory (Nielsen & Chuang, 2000; Wilde, 2017) and are used here as diagnostics rather than as substitutes for physical record certification. Record-relative coherence provides a further diagnostic. Relative to a declared completely positive, trace-preserving and idempotent dephasing or conditional-expectation map ΔRA, one may define CRA(ρA)=D ⁣(ρA∥ΔRA(ρA)). This is closely related to the relative-entropy formulation of quantum coherence (Baumgratz, Cramer & Plenio, 2014). A measurement frame need not be globally classical or highly mixed; what is required is that the degrees of freedom used as records possess the stability and coherence properties demanded by the declared protocol. The dissipative part of measurement is treated separately. On a faithful detailed-balance sector, quantum Markov dynamics admits entropy-production and gradient-flow structures associated with relative entropy (Spohn, 1978; Carlen & Maas, 2017). In the OT–GKSL architecture, this entropic contribution belongs to the native dynamical layer and is not identified with the full measurement cost. Hamiltonian, non-detailed-balance, record, calibration, bridge, detector, and inference contributions retain separate roles. Accordingly, the cost of a directed readout is protocol-dependent rather than a universal scalar entropy. Changes in mutual information or coherence may have either sign and are therefore treated as diagnostics unless an explicit physical conversion law is supplied. A complete readout burden is instead assembled from declared nonnegative resource terms and controlled remainders. This is compatible with the broader information-thermodynamic distinction between information, entropy production, work, erasure, measurement, and feedback costs (Landauer, 1961; Sagawa & Ueda, 2010). The relation A←B may therefore be viewed as a certified breaking of the prior status symmetry A↔B: before the record architecture is specified, neither subsystem is intrinsically the measuring reference. This is a symmetry of descriptive status, not a new dynamical symmetry of nature. The actual orientation is selected by the physical record structure and by the dependence of those records on the state-dependent content being measured. The paper also places ordinary measurement upstream of event and spacetime reconstruction. Measurement records, complete event records, event individuation, geometric coordinate records, and clock records are distinct structures. 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This paper provides a comprehensive analysis of the fundamental limits of quantum telescopingin experimental and noisy intermediate-scale quantum (NISQ) regimes. Building on thetheoretical telescoping framework developed in Parts I–IX, we rigorously study how physicalnoise, finite sampling, device drift, and imperfect control fundamentally cap achievable telescopingorder. We establish that in the presence of non-vanishing noise floors, all telescopingschemes satisfying our stated noise-and-measurement assumptions exhibit saturation beyond afinite refinement depth, regardless of their ideal convergence rate.We develop a complete mathematical framework incorporating:• Detailed noise models for NISQ devices with explicit error bounds• Measurement-induced lower bounds on observable telescoping increments• Statistical hypothesis testing framework for distinguishing telescoping orders• Rigorous conditions under which exponential telescoping becomes experimentally indistinguishableon NISQ devices with finite noise and polynomial resources• Device drift models and their impact on convergence verification• Quantitative trade-offs between circuit depth, sampling budget, and achievable precisionThese results formalize the unavoidable gap between asymptotic algorithmic guarantees andexperimentally realizable convergence, providing essential guidance for practical quantum algorithmdesign in the pre-fault-tolerant era.

•Feb 5, 2026•0 citations
We develop a composite operational architecture for sequential quantum measurements that (i) gives a tight bipartite order-effect bound with an explicit equality set characterized on the Halmos two-subspace block, (ii) upgrades Doeblin-type minorization to composite instruments and proves a product lower bound for the operational Doeblin constants, yielding data-driven exponential mixing rates, (iii) derives a diamond-norm commutator bound that quantifies how serial and parallel rearrangements influence observable deviations, and (iv) establishes a monitored Lindblad limit that links discrete look-return loops to continuous-time GKLS dynamics under transparent assumptions. Building on the GKLS framework of Gorini, Kossakowski, Sudarshan, Lindblad, Davies, Spohn, and later work of Fagnola-Rebolledo and Lami et al., we go beyond asymptotic statements by providing finite-sample certificates for the minorization parameter via exact binomial intervals and propagating them to rigorous bounds on the number of interaction steps required to attain a prescribed accuracy. A minimal qubit toy model and CSV-based scripts are supplied for full reproducibility. Our results position order-effect control and operational mixing on a single quantitative axis, from equality windows for pairs of projections to certified network mixing under monitoring. The framework targets readers in quantum information and quantum foundations who need explicit constants that are estimable from data and transferable to device-level guarantees.
We study sequential quantum measurements within an operational N–Q–S architecture and prove three main results. (i) A tight order-effect bound with saturation conditions characterized by Halmos two-subspace blocks; the construction extends to POVMs via Naimark dilation. (ii) A Doeblin-type minorization for instruments, yielding a quantitative 1-norm contraction and an exponential mixing rate γ ≥ −ln(1 − δ)/Δt for the look–return loop. (iii) A monitored discrete-to-GKLS bridge: as Δt → 0 under standard stability assumptions, the certified rate transfers to a compatible GKLS semigroup. We define testable indices (ONS/PII/PAD) linking our bounds to behavioral order effects, and we supply complete proofs (main text/appendices) together with a reproducibility package (code, data, and figure scripts).
Many tasks in open quantum systems and quantum information theory require guarantees on the mixing rate of a continuous-time Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) semigroup, while experimental access is typically limited to a discrete one-step completely positive trace-preserving (CPTP) map estimated from data. We provide an operational route from such one-step information to certified lower bounds on exponential mixing rates and to a compatible Lindblad generator. Given a statistically consistent estimate of a stationary channel E_delta t and one stationary state tau, we first certify a Doeblin minorization mass for E_delta t and translate it into a data-certified contraction rate gamma_op in trace distance. A Chernoff-type telescoping argument then controls the embedding error between discrete dynamics and any continuous-time realization, showing that the same rate holds up to an O(t Delta t) correction. Finally, we recover a GKLS generator by a convex semidefinite log-projection onto the generator cone, constrained to be consistent with the observed invariant state and the certificate. The resulting guarantee is non-asymptotic, depends only on observed counts and one stationary state, and bridges classical Doeblin theory with data-driven identification of quantum Markov semigroups from one-step CPTP experiments.
Pedagogical Guide, Foundations Audit, Uniqueness & Substitutability Analysis, Literature Comparison, and Reading Guidance : A Pedagogical Guide to Understanding the OT–GKSL Architecture This paper develops a relational axiomatics of quantum measurement within the Einstein-locked OT/GKSL source-readout architecture. Its central question is: under what physical conditions can one quantum system function as the measurement frame of another? The native dynamics is described on finite effective quantum supports by density operators and completely positive Markovian evolution of GKSL type (Gorini, Kossakowski & Sudarshan, 1976; Lindblad, 1976). At this level, the labels assigned to two subsystems do not by themselves determine which is the apparatus, the observer, or the measured object. This starting point is consonant with operational quantum measurement theory, in which measurements are represented by physical instruments, effects, and state transformations rather than by an unanalyzed external observer (Davies & Lewis, 1970; Ozawa, 1984), and with quantum-reference-frame approaches in which reference systems may themselves be finite quantum systems (Bartlett, Rudolph & Spekkens, 2007; Giacomini, Castro-Ruiz & Brukner, 2019). The specifically OT–GKSL proposal is that a measurement frame is a certified record-bearing subsystem. Given two finite systems A and B, the directed statement A←B means that state-dependent content associated with B is stably and physically encoded in records carried by A. The direction is not inferred from subsystem labels, entropy ordering, or correlation alone. It requires a declared record algebra or instrument, operationally accessible record channels, stability under the relevant open-system dynamics, sufficient distinguishability or identifiability of the encoded alternatives, and controlled native-to-readout bridge diagnostics. A measurement outcome label is not yet a physical record. For a quantum instrument {Iy}, a physical readout is represented by an explicit record register, for example MR(ρ)=∑yIy(ρ)⊗∣y⟩⟨y∣R, after which accessibility, persistence, distinguishability, and stability of the register can be tested. This distinction follows the operational treatment of quantum instruments (Davies & Lewis, 1970; Ozawa, 1984) while making the physical record requirement explicit inside the OT–GKSL architecture. The paper also clarifies the role of entropy in measurement. Neither S(A), S(B), nor the raw difference S(B)−S(A) determines the measurement direction. Mutual information I(A:B)=S(A)+S(B)−S(AB) diagnoses total correlation but is symmetric and therefore does not select a readout orientation. Quantum conditional entropy S(B∣A)=S(AB)−S(A) is relational but remains distinct from the classical conditional entropy obtained after a physical outcome register has formed, S(B∣Y)=∑ypyS(ρB∣y). These distinctions are standard in quantum information theory (Nielsen & Chuang, 2000; Wilde, 2017) and are used here as diagnostics rather than as substitutes for physical record certification. Record-relative coherence provides a further diagnostic. Relative to a declared completely positive, trace-preserving and idempotent dephasing or conditional-expectation map ΔRA, one may define CRA(ρA)=D ⁣(ρA∥ΔRA(ρA)). This is closely related to the relative-entropy formulation of quantum coherence (Baumgratz, Cramer & Plenio, 2014). A measurement frame need not be globally classical or highly mixed; what is required is that the degrees of freedom used as records possess the stability and coherence properties demanded by the declared protocol. The dissipative part of measurement is treated separately. On a faithful detailed-balance sector, quantum Markov dynamics admits entropy-production and gradient-flow structures associated with relative entropy (Spohn, 1978; Carlen & Maas, 2017). In the OT–GKSL architecture, this entropic contribution belongs to the native dynamical layer and is not identified with the full measurement cost. Hamiltonian, non-detailed-balance, record, calibration, bridge, detector, and inference contributions retain separate roles. Accordingly, the cost of a directed readout is protocol-dependent rather than a universal scalar entropy. Changes in mutual information or coherence may have either sign and are therefore treated as diagnostics unless an explicit physical conversion law is supplied. A complete readout burden is instead assembled from declared nonnegative resource terms and controlled remainders. This is compatible with the broader information-thermodynamic distinction between information, entropy production, work, erasure, measurement, and feedback costs (Landauer, 1961; Sagawa & Ueda, 2010). The relation A←B may therefore be viewed as a certified breaking of the prior status symmetry A↔B: before the record architecture is specified, neither subsystem is intrinsically the measuring reference. This is a symmetry of descriptive status, not a new dynamical symmetry of nature. The actual orientation is selected by the physical record structure and by the dependence of those records on the state-dependent content being measured. The paper also places ordinary measurement upstream of event and spacetime reconstruction. Measurement records, complete event records, event individuation, geometric coordinate records, and clock records are distinct structures. A record generated in a laboratory measurement does not by itself define a spacetime coordinate or a new event location. In the broader OT–GKSL architecture, complete certified records first support event individuation; a separate rank-D coordinate sector then supports geometric localization, with D=4 selected on the observed spacetime branch. The physical clock record is likewise distinct from both the native evolution parameter and the temporal coordinate used in the Lorentzian readout. This ordering connects the measurement axiomatics to the literature on decoherence and stable records, where environment-induced selection and redundant information storage explain how particular record structures can become effectively classical (Zurek, 2003; Ollivier, Poulin & Zurek, 2004), while leaving the subsequent event-to-geometry step as a specific feature of the OT–GKSL architecture. Finally, the paper formulates the associated Noether and balance-law structure. For a native observable Q, L†(Q)=0 implies conservation of its expectation value under the declared GKSL evolution. More general open-system symmetries and invariant sectors have been studied in Lindblad dynamics (Albert & Jiang, 2014). Dissipative entropy balances, however, are mathematically different objects from strict conserved charges. A further distinction arises at readout level: even when a native quantity is conserved, its physical record may contain calibration, representation, bridge, or support defects. Classical conservation is therefore a descended statement requiring both native conservation and controlled representation in the certified record channel. The resulting architecture provides an internal account of measurement without introducing a primitive observer. Reference frames, instruments, physical records, measured content, relational information, readout costs, and conservation laws are treated as typed and testable structures within the OT/GKSL source-readout architecture. Measurement then forms the upstream record layer from which the later OT–GKSL constructions of event individuation, calibrated clocks and frames, Lorentzian geometry, and classical spacetime readout can proceed. Bibliography: GKSL / Lindblad — foundational open-system framework for completely positive quantum dynamical semigroups. Carlen–Maas — bridge between quantum Markov semigroups, entropy production, and optimal transport geometry. Lovelock + Donoghue — Einstein-lock consistency and low-energy effective field theory (EFT) interpretation of gravity. Jacobson + Sakharov — gravity interpreted as an equation of state or induced/emergent phenomenon. Vassilevich / Seeley–DeWitt — spectral bridge from microscopic operators to geometry and effective actions. Bekenstein–Hawking–Wald — black-hole horizons, entropy, and Noether-charge formulations of gravitational thermodynamics. Wilson / Gross–Wilczek–Politzer — QCD, gauge structure, confinement, and asymptotic freedom. Kasevich–Chu / Peters–Chu / Rosi–Tino — atom-interferometric gravimetry and precision low-energy gravitational testing. Blais–Girvin–Oliver — transmon qubits and circuit-QED architectures relevant to CLCP/QBIT implementations. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ///Before reading: this document is a part of 20 documents that make up the full architecture. Each result presented here depends on those documents; links are provided below in this summary./// 1. Foundations of the Architecture: Foundations |GKSL/Lindblad ; Carlen–Maas ; Jacobson ; Sakharov ; Donoghue ; Lovelock) Establishes the core Einstein-locked OT/GKSL architecture for certified geometric readout and coherence-dependent gravitational sourcing. Optimal-Transport Gravity Trilemma | Identifies the certified operational boundary of geometric readout by proving the fundamental trade-off between temporal resolution, coframe stability, and bridge fidelity. Einstein–Readout Compatibility as a Certified Closure Criterion in the Einstein-Locked OT/GKSL Framework Technical Consolidation of Certified OT/GKSL Readout: Record Selection, Bridge Defects, OT Proxies, and Readout Calibration | Heat-Kernel Spectral Budgets and Entropic Transport in Einstein-Locked OT/GKSL Dynamics Fermionic Admissibility, Pauli Exclusion, and Creation–Annihilation Operators in the Einstein-Locked OT/GKSL Source–Readout Framework Quantum Measurement Without an Extern
This preprint develops an operational identifiability theory for bipartite quantum subsystem structure without presupposing an embedded tensor-product structure. It first proves a strict boundary between simultaneous-similarity information obtainable from self-consistent global process probabilities and the multiplication, adjoint, positivity, and order structure required for physical quantum subsystems. For finite-dimensional unitary return processes, the paper establishes a sharp zero/one/two-loop hierarchy. A tree leaves the full tensor-product-structure manifold unconstrained; one compatible return process leaves a continuous ambiguity of dimension (m-1)(n-1); and two Haar-generic compatible returns identify the common factor algebra up to exchange of equal-dimensional factors. From informationally complete probability tables, the construction M_s=G^{-1}P^{(s)} yields a calibration-free common-commutant obstruction. A finite-data singular-value certificate can rule out every nontrivial bipartition of the global dimension simultaneously. In the exact generic compatible case, the ranks of the primitive commutant idempotents also determine the previously unknown factor dimensions. After an independently justified physicality lift and explicit C^*-factor tests, the framework provides local reconstruction and finite-shot error guarantees. An exact two-qubit example demonstrates that two processes may each be factorizable in some tensor-product structure while admitting no common factorization.
We develop a quantitative framework for analyzing convergence and spectral-projector stability in finite-dimensional open quantum systems governed by Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) dynamics. Our focus is on computable bounds for the approach to the asymptotic sector and on perturbative stability estimates for the associated spectral projection. Rather than claiming universal constants for arbitrary Lindbladians, we formulate our results under explicit spectral-separation and resolvent assumptions, thereby making the role of non-normality transparent. We distinguish two geometrically different cases: (i) primitive semigroups with a unique steady state, and (ii) non-primitive semigroups with multidimensional steady-state structure relevant to encoded quantum information. We then illustrate the framework on truncated bosonic models motivated by dissipatively stabilized cat-qubit memories. This repository contains: (1) the preprint manuscript (v1.1) with Appendices A–C; (2) a Python validation suite implementing computation of resolvent norms, spectral gaps, projector stability, and primitivity detection; (3) 41 validated tests covering Propositions 1–2, Frigerio criterion, noise bias, and truncation error; (4) four numerical experiments on truncated bosonic models with publication-quality figures; (5) reproducible numerical results with checksums.
This paper provides a comprehensive analysis of the fundamental limits of quantum telescopingin experimental and noisy intermediate-scale quantum (NISQ) regimes. Building on thetheoretical telescoping framework developed in Parts I–IX, we rigorously study how physicalnoise, finite sampling, device drift, and imperfect control fundamentally cap achievable telescopingorder. We establish that in the presence of non-vanishing noise floors, all telescopingschemes satisfying our stated noise-and-measurement assumptions exhibit saturation beyond afinite refinement depth, regardless of their ideal convergence rate.We develop a complete mathematical framework incorporating:• Detailed noise models for NISQ devices with explicit error bounds• Measurement-induced lower bounds on observable telescoping increments• Statistical hypothesis testing framework for distinguishing telescoping orders• Rigorous conditions under which exponential telescoping becomes experimentally indistinguishableon NISQ devices with finite noise and polynomial resources• Device drift models and their impact on convergence verification• Quantitative trade-offs between circuit depth, sampling budget, and achievable precisionThese results formalize the unavoidable gap between asymptotic algorithmic guarantees andexperimentally realizable convergence, providing essential guidance for practical quantum algorithmdesign in the pre-fault-tolerant era.