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Reproducibility, Measurement Timing, and Structural Bias in a Discrete Probe-Geometry Backreaction Model

Paul Jarvis

Published Sep 14, 2026• 0 citations
Publisher

Abstract

We construct a minimal discrete model in which a quantum probe on a lattice is coupled to a single auxiliary two-level qubit representing local geometry, implemented three ways: continuous unitary evolution, unitary evolution interrupted by measurement on a fixed schedule, and unitary evolution interrupted by measurement triggered adaptively by the qubit's own entanglement entropy. We show continuous unitary evolution is bit-for-bit reproducible across runs, while both measurement-based implementations are reproducible only up to the first ambiguous measurement outcome, and the adaptive schedule introduces a further failure mode: runs can diverge in when subsequent measurements occur, not just their outcome. Using GPU-batched Monte Carlo ensembles of 200,000 trajectories, we confirm both effects are structural, not artifacts of particular seed choices (77.5% and 73% convergence rates respectively), and give an analytic argument for both, checked directly against exact simulation at every measurement window. The work does not resolve which class of backreaction describes any real physical system, but provides a concrete, checkable diagnostic for backreaction models.

DOI
10.5281/zenodo.22753640

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We construct a minimal discrete model in which a quantum probe on a lattice is coupled to a single auxiliary two-level qubit representing local geometry, implemented three ways: continuous unitary evolution, unitary evolution interrupted by measurement on a fixed schedule, and unitary evolution interrupted by measurement triggered adaptively by the qubit's own entanglement entropy. We show continuous unitary evolution is bit-for-bit reproducible across runs, while both measurement-based implementations are reproducible only up to the first ambiguous measurement outcome, and the adaptive schedule introduces a further failure mode: runs can diverge in when subsequent measurements occur, not just their outcome. Using GPU-batched Monte Carlo ensembles of 200,000 trajectories, we confirm both effects are structural, not artifacts of particular seed choices (77.5% and 73% convergence rates respectively), and give an analytic argument for both, checked directly against exact simulation at every measurement window. The work does not resolve which class of backreaction describes any real physical system, but provides a concrete, checkable diagnostic for backreaction models.
Measurements on a quantum particle unavoidably affect its state, since the otherwise unitary evolution of the system is interrupted by a nonunitary projection operation. To probe measurement-induced effects in the state dynamics using a quantum simulator, the challenge is to implement controlled measurements on a small subspace of the system and continue the evolution from the complementary subspace. A powerful platform for versatile quantum evolution is represented by photonic quantum walks because of their high control over all relevant parameters. However, measurement-induced dynamics in such a platform have not yet been realized. We implement controlled measurements in a discrete-time quantum walk based on time-multiplexing. This is achieved by adding a deterministic outcoupling of the optical signal to include measurements constrained to specific positions resulting in the projection of the walker's state on the remaining ones. With this platform and coherent input light, we experimentally simulate measurement-induced single-particle quantum dynamics. We demonstrate the difference between dynamics with only a single measurement at the final step and those including measurements during the evolution. To this aim, we study recurrence as a figure of merit, that is, the return probability to the walker's starting position, which is measured in the two cases. We track the development of the return probability over 36 time steps and observe the onset of both recurrent and transient evolution as an effect of the different measurement schemes, a signature which only emerges for quantum systems. Our simulation of the observed one-particle conditional quantum dynamics does not require a genuine quantum particle but is demonstrated with coherent light.
Starting from a state of low quantum entanglement, local unitary time evolution increases the entanglement of a quantum many-body system. In contrast, local projective measurements disentangle degrees of freedom and decrease entanglement. We study the interplay of these competing tendencies by considering time evolution combining both unitary and projective dynamics. We begin by constructing a toy model of Bell pair dynamics which demonstrates that measurements can keep a system in a state of low, i.e., area-law, entanglement, in contrast with the volume-law entanglement produced by generic pure unitary time evolution. While the simplest Bell pair model has area-law entanglement for any measurement rate, as seen in certain noninteracting systems, we show that more generic models of entanglement can feature an area-to-volume law transition at a critical value of the measurement rate, in agreement with recent numerical investigations. As a concrete example of these ideas, we analytically investigate Clifford evolution in qubit systems which can exhibit an entanglement transition. We are able to identify stabilizer size distributions characterizing the area law, volume law, and critical ``fixed points.'' We also discuss a Floquet random unitary circuit, where the answers depend on the order of limits---one order of limits yields area-law entanglement for any nonzero measurement rate, whereas a different order of limits allows for an arealaw--volumelaw transition. Finally, we provide a rigorous argument that a system subjected to projective measurements can only exhibit a volume-law entanglement entropy if it also features a subleading correction term, which provides a universal signature of projective dynamics in the high-entanglement phase.
This work proposes a minimal dynamical completion of quantum mechanics that unifies continuous unitary evolution with definite single-run outcomes. While decoherence explains why certain measurement bases become stable, it does not explain why individual experiments yield one specific result. Here, measurement context selects a preferred observable A(Con)A(\mathrm{Con})A(Con), and a context-dependent stochastic extension of Schrödinger evolution—with strength κ(Con)\kappa(\mathrm{Con})κ(Con)—drives almost-sure localization onto its eigenspaces. The stochastic trajectories reproduce Born probabilities as hitting probabilities, reduce to standard unitary dynamics when κ(Con)→0\kappa(\mathrm{Con}) \to 0κ(Con)→0, and yield an ensemble master equation of Lindblad form. Framed operationally via CPTP instruments, the model supports both epistemic and dynamical interpretations, offering an interpretation-neutral account of single-run definiteness fully consistent with standard quantum mechanics in the appropriate limit.
This bundle contains the three Python scripts and supporting output files used to produce every numerical result in the companion paper "Propagation, Closure, and the Born Rule: An ENSO Derivation in the Two-State Case, Verified Numerically and Extended to Three States" (Needham, 2026, v8). The bundle has two distinct numerical layers, kept separate throughout the paper and throughout this archive. Layer A (theorem verification): The script validate_v7.py is a self-contained validator of the closure-law derivation. It checks 19 independent numerical and algebraic claims across six blocks: the ENSO constants (phi, RG, Z0, Delta_tau) from the Boundary Shell paper; the algebraic identities used in the Itô-norm-preservation derivation of the QSD coefficient ratio; the catastrophic 60x failure of a two-rate counterexample; the Born tracking of the derived single-rate dynamics at N = 10,000 trajectories below the Monte Carlo noise floor; the three-state extension of the result to C = diag(1, 0, -1); and the gamma-independence of Born statistics under variation of the apparatus coupling lambda. Exit code 0 on full pass. Total runtime under ten minutes on a standard laptop. Layer B (realization phenomenology): The scripts compare_schrodinger_enso_reference.py and scan_event_time_distributions.py implement an explicit propagation-plus-threshold scenario in which the closure dynamics runs alongside plain unitary Schrödinger evolution and a standard continuous-measurement reference model on the same Wiener-increment seeds. The first script compares outcome frequencies and event times across initial probabilities; the second scans the apparatus coupling lambda from 10 to 100 and measures the two-sample Kolmogorov-Smirnov distance between the ENSO and reference event-time distributions at each coupling. The principal phenomenological finding is that outcome frequencies remain broadly stable across the scan while event-time distributions separate progressively, with KS distance reaching approximately 0.99 at lambda = 100 for edge probabilities. The paper proposes this timing separation as a candidate experimental discriminator between ENSO closure and standard collapse in circuit-QED continuous-measurement experiments at variable measurement strength. Output CSVs and PNGs from both layers are included for reference. The validator can be run as: python3 validate_v7.pypython3 validate_v7.py --quickpython3 validate_v7.py --no-plot The Layer B scripts can be run as: python3 compare_schrodinger_enso_reference.pypython3 scan_event_time_distributions.py Dependencies: Python >= 3.10, NumPy, Matplotlib. CPU only. No GPU required. No other external dependencies. All seeds are deterministic; bundled CSV files are reproducible bit-for-bit modulo platform-dependent floating-point accumulation that does not affect the summary statistics reported in the paper. This bundle is uploaded to Zenodo for archival timestamping and reproducibility, and to provide the readership of the paper with direct access to the verification machinery. See the included README.md for a full file-by-file inventory and a table mapping each numerical claim in the paper to the script and block that reproduce it. For Further Information abou the ENSO Framework, please contact, Eric Needham:ensotheory1@gmail.com
The backaction of quantum measurements is traditionally regarded as an unavoidable source of noise that limits the precision of state estimation and control. In this work we propose a novel conceptual framework, the Quantum Measurement Backaction Topology (QMBT), in which the measurement backaction is understood as possessing a topological structure. We introduce the notion of a measurement backaction manifold T_M, defined by the set of all possible dynamical trajectories induced by continuous measurements. By mapping the evolution of the density matrix under measurement to a geometrical flow on T_M, we identify conserved quantities and topological invariants that constrain the backaction dynamics. The QMBT formalism is formulated using stochastic master equations with measurement operators M_k and corresponding noise terms, and we show that the resulting equations can be cast into a form reminiscent of gauge theories on fiber bundles. This perspective allows us to classify measurement protocols according to their topological class and to predict robustness against parameter variations. We illustrate the theory with continuous monitoring of a two–level system and a bosonic mode, demonstrating how the topology dictates the long–time behavior of the system and the achievable measurement precision. The QMBT framework opens a pathway to modify quantum measurement theory fundamentally, potentially enabling the design of measurement schemes that evade the traditional noise floor and thereby unlocking new regimes of quantum control and metrology.
We illustrate a reverse Von Neumann measurement scheme in which a geometric phase induced on a quantum harmonic oscillator is measured using a microscopic qubit as a probe. We show how such a phase, generated by a cyclic evolution in the phase space of the harmonic oscillator, can be kicked back on the qubit, which plays the role of a quantum interferometer. We also extend our study to finite-temperature dissipative Markovian dynamics and discuss potential implementations in micro- and nanomechanical devices coupled to an effective two-level system.