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

Kazuyuki Yoshida

TL;DR

The paper addresses how to quantify order effects and mixing in sequential quantum measurements using a bipartite, compositional framework (N--Q--S). It develops explicit, data-driven constants for order-induced deviations, minorization-based mixing rates for composite instruments, and a diamond-norm bound that ties rearrangements to observable deviations, while connecting discrete look–return loops to a monitored Lindblad (GKLS) limit with finite-sample guarantees. Key contributions include a tight Halmos two-subspace bound with an explicit equality characterization, a product Doeblin constant bound yielding nonasymptotic mixing rates, and a monitored dynamics limit that preserves device-level rates. The results are implemented with a minimal qubit toy model and CSV scripts to support reproducibility, enabling device-level guarantees and data-driven performance certificates for order-effect control and operational mixing in quantum information tasks.

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.

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

TL;DR

The paper addresses how to quantify order effects and mixing in sequential quantum measurements using a bipartite, compositional framework (N--Q--S). It develops explicit, data-driven constants for order-induced deviations, minorization-based mixing rates for composite instruments, and a diamond-norm bound that ties rearrangements to observable deviations, while connecting discrete look–return loops to a monitored Lindblad (GKLS) limit with finite-sample guarantees. Key contributions include a tight Halmos two-subspace bound with an explicit equality characterization, a product Doeblin constant bound yielding nonasymptotic mixing rates, and a monitored dynamics limit that preserves device-level rates. The results are implemented with a minimal qubit toy model and CSV scripts to support reproducibility, enabling device-level guarantees and data-driven performance certificates for order-effect control and operational mixing in quantum information tasks.

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.

Paper Structure

This paper contains 25 sections, 5 theorems, 24 equations, 2 figures, 2 tables.

Table of Contents

  1. Note on terminology.
  2. Introduction
  3. Related work
  4. Order effects, projection geometry, and equality characterizations
  5. Doeblin/Dobrushin minorization, channel contraction, and mixing rates
  6. Monitored dynamics, quantum trajectories, and Lindblad limits
  7. Combs, process matrices, and causal structure
  8. Positioning and limitations.
  9. Standing assumptions.
  10. Background and methods
  11. N--Q--S layers and notation
  12. Channels, instruments, and tensor products
  13. Results
  14. Estimating the Doeblin constant and a nonasymptotic mixing bound
  15. Frequency estimator.
  16. ...and 10 more sections

Key Result

Theorem 4.4

Let $P,Q$ be orthogonal projections on a finite-dimensional Hilbert space. Restrict to a Halmos two-subspace block with principal angle $\theta\in[0,\pi/2]$ and write the commutator in the canonical form $[P,Q]=\tfrac{1}{2}\sin 2\theta\, J$ with $J=\left(\right)$ on that block. For any unit vector $ and the bipartite order-effect bound is tight: Moreover, the following are equivalent: Equality i

Figures (2)

  • Figure 1: Trend of the bipartite order-effect proxy under partial $ZZ$ coupling. (uniform grid over instrument parameters; fixed random seed; script and exact settings to be provided via Zenodo, DOI:10.5281/zenodo.17959208). (For devices, $\gamma$ is instantiated via calibrated $H_{\mathrm{int}}$ and duration $t$ as detailed in the short subsection "From simulation to device", and the plotted proxy is computed from experimentally accessible order-difference probabilities.) The growth with coupling strength $\gamma$ (mean/min/max over $(\alpha,\beta)$ grid) visualizes the tightness trend of the bipartite bound and provides a reproducibility baseline for the qubit toy model.
  • Figure 2: Halmos block and equality window. Left: two one-dimensional ranges with principal angle $\theta$ (Halmos 2D block). Right: pure states in the block (Bloch circle). The vertical red diameter marks states aligned with eigenvectors of $J$, where $|\langle\psi|[P,Q]|\psi\rangle|$ attains its maximum; the overall scale is $\tfrac12|\sin 2\theta|$.

Theorems & Definitions (13)

  • Definition 4.1: Composite N--Q--S
  • Definition 4.2: Serial and parallel composition
  • Remark 4.3: Non--signalling desideratum
  • Theorem 4.4: Equality window on a Halmos block
  • proof : Proof sketch
  • Example 4.5: Product Doeblin constant for dephasing channels
  • Lemma 4.6: Product minorization
  • proof
  • Lemma 4.7: Trace-norm contraction on traceless part
  • Remark 4.8: Primitive case and fixed-point axis
  • ...and 3 more