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arXiv:2005.03886 · Linear Algebra and its Applications

A non-commutative Bayes' theorem

Arthur J. Parzygnat, Benjamin P. Russo

Published Feb 28, 2022• 24 citations
arXivHTMLPDF
DOI
10.1016/j.laa.2022.02.030

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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)