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Quantum Data Statistics: An Operator-Theoretic Framework for Structured Uncertainty

Pedro Caceres

Published Jan 2, 2026• 0 citations
Publisher

Abstract

This repository contains Python code implementing numerical simulations for the paper "Quantum Data Statistics: An Operator-Theoretic Framework for Structured Uncertainty". The code demonstrates how Quantum Data Statistics (QDS) — a novel operator-valued statistical framework — can detect and characterize bifurcations, chaos, and structural transitions in complex dynamical systems more sensitively than classical moment-based statistics.

DOI
10.5281/zenodo.18122366

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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)

•May 8, 2026•0 citations
This repository contains Python code implementing numerical simulations for the paper "Quantum Data Statistics: An Operator-Theoretic Framework for Structured Uncertainty". The code demonstrates how Quantum Data Statistics (QDS) — a novel operator-valued statistical framework — can detect and characterize bifurcations, chaos, and structural transitions in complex dynamical systems more sensitively than classical moment-based statistics.
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.
This repository contains the full dataset and analysis code accompanying the manuscript“Emergent AdS₂ Geometry from Quantum Codes: Experimental Evidence for the Fion Substrate Framework.” Using a 5-qubit ] stabilizer code implemented on IBM Quantum hardware, we reconstruct an effective information-geometric metric from 12,000 measurement shots. Bias-corrected mutual information, hyperbolic and Euclidean embeddings, stress minimization and Akaike Information Criterion (AIC) model comparison all point to a hyperbolic AdS₂ geometry with curvatureκ = −1.0012 ± 0.0039 (90% BCa CI), statistically equivalent to the theoretical value κ = −1 (TOST equivalence test, ΔAIC = 30.32 in favour of the AdS₂ model). In the associated manuscript we interpret these results within the “fion substrate framework”, a unified informational model in which elementary quantum codes generate spacetime, matter and (at high coherence) mind-like dynamics. The Figshare materials, however, focus on the quantum-information and geometric side of the work. Contents of the repository raw IBM Quantum measurement data (counts) for the 5-qubit code (12,000 shots); processed mutual-information matrices and distance matrices; Python scripts for metric embedding, bootstrap uncertainty estimation and equivalence testing (BCa + TOST); bootstrap distributions of the curvature κ and model-comparison summaries (AIC, stress); documentation on how to reproduce all numerical results and figures reported in the paper. The dataset is intended to allow full reproduction and independent re-analysis of the emergent-geometry reconstruction, and to provide a starting point for extending the protocol to larger quantum codes and higher-dimensional AdS geometries
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)