Tracking the Oscillations of a Single Qubit by Means of Sequential Measurements

Vincenzo Matta, Vincenzo Claudio Pierro

A one-dimensional quantum oscillator is monitored by taking repeated position measurements. As a first contribution, it is shown that, under a quantum nondemolition measurement scheme applied to a system initially at the ground state, (i) the observed sequence of measurements (quantum tracks) corresponding to a single experiment converges to a limit point, and that (ii) the limit point is random over the ensemble of the experiments, being distributed as a zero-mean Gaussian random variable with a variance at most equal to the ground-state variance. As a second contribution, the richer scenario where the oscillator is coupled with a frozen (i.e., at the ground state) ensemble of independent quantum oscillators is considered. A sharply different behavior emerges: under the same measurement scheme, here we observe that the measurement sequences are essentially divergent. Such a rigorous statistical analysis of the sequential measurement process might be useful for characterizing the main quantities that are currently used for inference, manipulation, and monitoring of many quantum systems. Several interesting properties of the quantum tracks evolution, as well as of the associated (quantum) threshold crossing times, are discussed and the dependence upon the main system parameters (e.g., the choice of the measurement sampling time, the degree of interaction with the environment, the measurement device accuracy) is elucidated. At a more fundamental level, it is seen that, as an application of basic quantum mechanics principles, a sharp difference exists between the intrinsic randomness unavoidably present in any quantum system, and the extrinsic randomness arising from the environmental coupling, i.e., the randomness induced by an external source of disturbance.

Jason F. Ralph, Joshua Combes, Howard M. Wiseman

Hermann Uys, Pieter du Toit, S. C. Burd +2 more

Qing Xu, A.A. Mondragon, Philippe B. Gallion +1 more

We present a homodyne receiver structure for the detection of weak coherent states that uses sequential in-phase and quadrature measurements on the received optical signal. This receiver performs the optical carrier phase tracking requiring only a single balanced homodyne detector, by including a postdetection Costas-loop-type feedback, which additionally allows the use of suppressed carrier modulations in the received field, for efficient transmission. We report an experimental interferometric self-homodyne setup for the sequential detection of low photon number, binary phase-modulated optical signals that consist of strongly attenuated laser pulses by using a reference field as the local oscillator with an alternatively switched phase. A Costas loop postdetection subsystem is implemented in discrete time to perform fast real-time optical phase tracking. We also present the experimental results of the homodyne postdetection statistics for received BPSK signals with very low photon numbers, and compare them with the theoretical uncertainty limit. Finally, we conduct bit error rate measurements over a wide range of signal level, as well as a comparison with the standard quantum limit.

Koray Mentesoglu, Rahul Trivedi, Sara Mouradian

We present a protocol for simultaneous estimation of both quadratures of a displacement channel on a quantum harmonic oscillator using a single ancilla qubit, without complex state preparation or adaptive control. The qubit is coupled to the quantum harmonic oscillator via controlled displacements of strength $A$ and undergoes $N$ rounds of measurement and reset, while the oscillator coherently accumulates the displacement signal. The non-commuting displacements imprint an order-dependent geometric phase at each round, giving a quantum Fisher information that scales as $\mathcal{O}(A^{2} N^{3})$. Although the measurement record space grows as $2^N$, we show that exact likelihoods for any individual measurement record can be computed in $\mathcal{O}(N^{2})$ time. With this, we show that measurement in the computational basis approximately saturates the quantum bound and maintains the $\mathcal{O}(A^2 N^3)$ scaling. Furthermore, the periodic information extraction makes the protocol robust to decoherence. Our results provide a framework for the design and analysis of sequential measurement in ancilla-assisted protocols.

Miguel Percudani

This record contains the complete development of a two-mode Jaynes–Cummings model of endogenous and exogenous phase coherence, organized in two independent tracks. TRACK A — FORMAL AND FALSIFIABLE A two-mode quantum model in which an analytic sequential mode (a qubit) interacts with a non-analytic substrate mode (a truncated harmonic oscillator) through a time-dependent Jaynes–Cummings coupling with Lindblad dissipation. The coupling is modulated by an empirically accessible signal identified with Default Mode Network (DMN) activity. The model contains no speculative constants: all parameters are free and listed in the main paper. Numerical integration with an implicit BDF solver produces a two-dimensional phase diagram in the plane (α = g/δ, γφ), where the boundary between analytic-dominated and substrate-dominated steady states is a curve rather than a single threshold. The paper states four falsifiable predictions testable with current EEG/fMRI techniques. Track A documents: • track_A_formal.pdf — main paper, 7 pages • track_A_field_equations.pdf — technical appendix, 15 pages • track_A_Extended_version_full_derivation.pdf — extended appendix, 20 pages, supersedes the previous one and adds full derivations of the mean-field closure, the dressed-state expansion, and the linear stability of the Liouvillian superoperator TRACK B — SPECULATIVE RESEARCH PROGRAM An ontological interpretation of the same two-mode structure in terms of the Binary Universe Ontology: Bit 1 = analytic qubit (baryonic, finite, sequential), Bit 0 = substrate oscillator (non-baryonic, infinite, superposed); Jaynes–Cummings exchange = causal friction; Lindblad dissipation = entropic homeostasis; DMN = baryonic filter. The document presents the geometric constants Φ_UAT = π²/8, kUAT ≈ 10.90677, and κcrit ≈ 10⁻⁷⁸ explicitly as speculative hypotheses, not as derived results. Track B documents: • track_B_speculative.pdf — 6 pages INDEPENDENCE OF THE TWO TRACKS Track A does not use any constant from Track B. Track B does not modify the numerical results of Track A. Track A remains valid even if Track B is refuted. RECORD CONTENTS • 4 PDF documents (main paper, two technical appendices, speculative interpretation) • 4 LaTeX sources, compilable with pdflatex • 3 standalone figures in PNG format (Rabi oscillation, 2D phase diagram, 1D cuts) • 6 code archives (trackA_v2 through trackA_v6) documenting the full historical development, with v5 and v6 being the versions that generated the published results NOTES The 20 fMRI sessions of Agustina Vélez Picatto (led by Pablo Barttfeld, IIPsi — CONICET / Universidad Nacional de Córdoba) are cited as compatibility, not validation, as of 22 September 2026. No peer-reviewed DOI has been issued yet. Reproducibility: all results can be regenerated from trackA_v6.zip using standard Python scientific libraries (numpy, scipy, pandas, matplotlib) and the BDF implicit solver with rtol = 1e-7, atol = 1e-9. Compilation of the LaTeX sources requires two pdflatex passes for cross-references. License: CC BY 4.0 for manuscripts, MIT for code.