How to construct the optimal Bayesian measurement in quantum statistical decision theory

Masao Ozawa

IHSIEN CHOU

The many-worlds interpretation of quantum mechanics by Hugh Everett III Proposed in 1957, this framework argues that the universe branches into multiple parallel realities during quantum measurement, thus avoiding the wave function collapse hypothesis. This framework has profound philosophical and practical implications, particularly in understanding decision-making processes under uncertainty. While observers cannot directly select or optimize specific quantum branches , within the framework of decision theory and quantum Bayesianism , they can optimize the distribution of expected outcomes through strategic decision-making . This involves the complex interaction between probability weights and subjective experience in the many-worlds interpretation.

Alexander Semenovich Holevo

Fuyuhiko Tanaka

Recently, many fundamental and important results in statistical decision theory have been extended to the quantum system. Quantum Hunt-Stein theorem and quantum locally asymptotic normality are typical successful examples. In the present paper, we show quantum minimax theorem, which is also an extension of a well-known result, minimax theorem in statistical decision theory, first shown by Wald and generalized by LeCam. Our assertions hold for every closed convex set of measurements and for general parametric models of density operator. On the other hand, Bayesian analysis based on least favorable priors has been widely used in classical statistics and is expected to play a crucial role in quantum statistics. According to this trend, we also show the existence of least favorable priors, which seems to be new even in classical statistics.

Itamar Pitowsky

Richard N. Silver

. Can quantum probability theory be applied, beyond the microscopic scale of atoms and quarks, to the human problem of reasoning from incomplete and uncertain data? A unified theory of quantum statistical mechanics and Bayesian statistical inference is proposed. QSI is applied to ordinary data analysis problems such as the interpolation and deconvolution of continuous density functions from both exact and noisy data. The information measure has a classical limit of negative entropy and a quantum limit of Fisher information (kinetic energy). A smoothing parameter analogous to a de Broglie wavelength is determined by Bayesian methods. There is no statistical regularization parameter. A priori criteria are developed for good and bad measurements in an experimental design. The optimal image is estimated along with statistical and incompleteness errors. QSI yields significantly better images than the maximum entropy method, because it explicitly accounts for image continuity. Introduction Jaynes has been an eloquent advocate for a compelling hypothesis: Probability Theory as Logic . That is, probabilities represent degrees of belief; probability theory develops and applies universal principles of logical inference from incomplete information. Two of his primary interests have been Bayesian probability theory and the interpretation of quantum mechanics. Bayesian probability theory yields inferences by systematically and consistently combining new data with prior knowledge. Jaynes pioneered the maximum entropy (ME) class of Bayesian methods for density function estimation (Jaynes, 1983), which has been applied successfully to numerous data analysis problems.