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QUANTUM DATA STATISTICS: AN OPERATOR-THEORETIC FRAMEWORK FOR STRUCTURED UNCERTAINTY

Zeraoulia Rafik

Published Jan 13, 2026• 0 citations
OpenAlex

Abstract

We introduce Quantum Data Statistics (QDS), an operator-theoretic framework for representing structured uncertainty in time-evolving empirical systems. Empirical probability distributions are embedded into a Hilbert space via a square-root amplitude mapping and represented as density operators, enabling geometric analysis of uncertainty beyond scalar summaries. QDS states admit spectral decomposition where eigenvalues quantify regime weights and eigenvectors capture latent modes; coherences encode interactions between them. Classical statistics emerge as the diagonal subcase, but QDS reveals off-diagonal structure invisible to moments. Key invariants-purity P, entropy S, coherence C, and dual uncertainty U -detect bifurcations and chaos more sensitively than variance or Lyapunov exponents, as validated on a quantum logistic map: sharp transitions at r ≈ 3.0, 3.45, 3.57; early instability warnings; and separable phase portraits in (P, S, U )-space. This work founds QDS mathematically; inference and applications follow in future studies.

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The information of ambiguity

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The phase space characteristics of a quantum state are best captured by the Wigner distribution. This displays transparently the diagonality information of the density matrix. The complementary function offering transparently the off-diagonal elements is captured by a function called the S-function, or the ambiguity. In carrying the maximal information about the quantum coherences it represents the uncertainties or ambiguity of the diagonal information. Mathematically this is manifested in its role as the phase space moment generating function. Formally it complements the information in the Wigner function. These formal relations provide the starting point for the present investigations. As a measure of quantum uncertainties, ambiguity may be used to define a probability measure on the off-diagonality. The mathematical and physical consistency of this view is presented in this paper. For a pure state, we find the extraordinary result that such distributions are their own Fourier transforms. The physical interpretation of this distribution as a carrier of classical signal fuzziness suggests the introduction of heuristic approximations to the observational uncertainties. We illustrate the properties and interpretation of the ambiguity function by some specific examples. We find that for smooth, ‘Gaussian-like’ distributions, the heuristic considerations provide good approximations. On the other hand, representing quantum interferences, the ambiguity serves as the most positive probe for the ultimate quantum structures which have been called sub-Planckian. They are interesting because it has been argued that such structures are physically observable.

Journal of Modern Optics•Jun 10, 2009•1 citation
We introduce Quantum Data Statistics (QDS), an operator-theoretic framework for representing structured uncertainty in time-evolving empirical systems. Empirical probability distributions are embedded into a Hilbert space via a square-root amplitude mapping and represented as density operators, enabling geometric analysis of uncertainty beyond scalar summaries. QDS states admit spectral decomposition where eigenvalues quantify regime weights and eigenvectors capture latent modes; coherences encode interactions between them. Classical statistics emerge as the diagonal subcase, but QDS reveals off-diagonal structure invisible to moments. Key invariants-purity P, entropy S, coherence C, and dual uncertainty U -detect bifurcations and chaos more sensitively than variance or Lyapunov exponents, as validated on a quantum logistic map: sharp transitions at r ≈ 3.0, 3.45, 3.57; early instability warnings; and separable phase portraits in (P, S, U )-space. This work founds QDS mathematically; inference and applications follow in future studies.
Quantum Data Statistics (QDS) replaces classical, commutative summaries of uncertainty by operatorvalued statistical states. A rolling empirical window in a financial time series defines an empirical distribution,which is lifted to an amplitude in a Hilbert space and then represented as a density operator. This representationpreserves mixture, interaction (coherence), and eigenvector geometry—features that are systematically lostunder classical projection to scalar indicators or diagonal statistics. This paper develops a publishable research-style foundation for QDS in finance with three new components:(i) a fully specified operator-valued Bayesian learning model and decision theory on the convex state space ofdensity operators; (ii) a family of QDS test statistics for structural change and regime detection in operatortime series, including asymptotic distribution theory for Bures/relative-entropy tests and a QDS-CUSUMprocedure; and (iii) a bifurcation analysis of QDS tests and readiness rules under parametric admissibledynamics (CPTP maps), yielding early-warning criteria in terms of superoperator spectra.We prove: a quantum Bayes update theorem for general instruments; a consistency theorem connectingHellinger convergence of empirical densities to trace-norm convergence of QDS posteriors; a contraction-basedstationarity theorem for operator autoregressive dynamics; and a risk-gap dominance theorem showing thatclassical-shadow Bayes rules are strict quotients of QDS Bayes rules and generically incur larger Bayes riskwhenever losses depend on noncommutative structure. Finally, we provide a detailed blueprint for enterprisedeployment (treasury risk and liquidity, institutional execution, marketing demand signals) and for cryptofinance (cross-exchange microstructure, stablecoin stress, and regime coupling)
Quantum Causal Algebra for Data Representation introduces a new mathematical framework that replaces statistical correlation with quantum causal influence as the foundation of representation learning. Classical data analysis relies on symmetric covariance or correlation matrices, but classical causality is fundamentally directional and incompatible with spectral methods such as PCA. This paper proposes the Quantum Causal Matrix (QCM)—a Hermitian operator encoding causal influence through quantum process matrices, entanglement, and indefinite causal order. We further define the Quantum Causal Tensor (QCT) for higher-order causal structures and develop Quantum Causal PCA (QC-PCA) as a generalization of classical PCA in Hilbert space. This work establishes a novel operator-theoretic and geometric foundation for causal data analysis, bridging quantum information, tensor algebra, and machine learning.
Quantum decoherence is both the fundamental mechanism underlying the quantum-to-classical transition and a major challenge for the development of scalable nanoscale quantum technologies. This work introduces a Quantum Phase Space (QPS) framework that provides a unified geometric description of decoherence. The framework is based on a dual structure consisting of an overcomplete continuous frame of minimum-uncertainty states that defines the QPS geometry, and an orthonormal basis that satisfies the strict mathematical requirements for decoherence within the Spectrum Broadcast Structure (SBS) objectivity criterion for pointer states. Within this framework, the variance-covariance matrix of the QPS ground states serves as a universal indicator of decoherence regimes:it is time-independent for Markovian (memoryless) dynamics, and time-dependent for non-Markovian dynamics with memory and information backflow. To illustrate the formalism, the Hu-Paz-Zhang (HPZ) model is generalized to include simultaneous position and momentum couplings to the environment, leading to a generalized non-Markovian master equation characterized by a spectral-density matrix. The corresponding evolution equations for the first moments (⟨p(t)⟩,⟨x(t)⟩)\ (\left\langle p(t) \right\rangle,\left\langle x(t) \right\rangle) (⟨p(t)⟩,⟨x(t)⟩) and for the ground covariance matrix establish a direct connection between microscopic environmental properties, classical-like trajectories and the QPS geometry. The proposed framework provides a unified theoretical foundation for modeling decoherence in nanoscale systems involving simultaneous position and momentum interactions with the environment. The QPS framework may thus bridge fundamental theory and practical quantum engineering, offering a promising coherent pathway to understand, control, and exploit decoherence at the nanoscience frontier.
The dynamical structure of quantum mechanics is conventionally formulated through Hamiltoniandriven unitary evolution, while entropy is typically treated as a statistical descriptor associatedwith measurement outcomes or environmental interactions. In this work, we propose the QuantumCausal Entropy (QCE) framework, an entropy-driven extension of quantum dynamics in whichentropy-weighted causal operators contribute directly to the evolution of quantum states. The proposed formulation introduces an entropy-dependent correction to the dynamical generator whilepreserving Hermiticity, probability conservation, and the Hilbert-space structure of conventionalquantum theory. Within this approach, standard Schr¨odinger dynamics emerge as a limiting casecorresponding to vanishing entropy-coupling strength, ensuring consistency with established experimental results. The framework predicts entropy-dependent modifications to transition amplitudes, decoherence rates, and state-selection probabilities in systems exhibiting significant entropygradients in state space. Possible experimental signatures may arise in high-coherence quantumplatforms, including superconducting qubits, atomic interferometers, and quantum optical systems.By incorporating entropy as an explicit causal element of quantum evolution, the QCE formulation provides a unified information-theoretic perspective linking quantum dynamics, decoherenceprocesses, and entropy-driven information flow, offering a foundation for future theoretical andexperimental investigations of entropy-mediated quantum phenomena.
The phase space characteristics of a quantum state are best captured by the Wigner distribution. This displays transparently the diagonality information of the density matrix. The complementary function offering transparently the off-diagonal elements is captured by a function called the S-function, or the ambiguity. In carrying the maximal information about the quantum coherences it represents the uncertainties or ambiguity of the diagonal information. Mathematically this is manifested in its role as the phase space moment generating function. Formally it complements the information in the Wigner function. These formal relations provide the starting point for the present investigations. As a measure of quantum uncertainties, ambiguity may be used to define a probability measure on the off-diagonality. The mathematical and physical consistency of this view is presented in this paper. For a pure state, we find the extraordinary result that such distributions are their own Fourier transforms. The physical interpretation of this distribution as a carrier of classical signal fuzziness suggests the introduction of heuristic approximations to the observational uncertainties. We illustrate the properties and interpretation of the ambiguity function by some specific examples. We find that for smooth, ‘Gaussian-like’ distributions, the heuristic considerations provide good approximations. On the other hand, representing quantum interferences, the ambiguity serves as the most positive probe for the ultimate quantum structures which have been called sub-Planckian. They are interesting because it has been argued that such structures are physically observable.