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Kluwer Academic Publishers eBooks

The Asymptotic Efficiency of the Consistent Estimator, Berry-Uhlmann’s Curvature, and Quantum Information Geometry

Keiji Matsumoto

Published Dec 19, 2005• 3 citations
Publisher

Abstract

It is pointed out that Berry-Uhlmann’s parallelism plays key role in the first order asymptotic theory of quantum statistical estimation, and that the parallelism is closely related to Nagaoka’s quantum information geometry, which is already successfully applied to quantum estimation theory. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/0-306-47097-7_14

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It is pointed out that Berry-Uhlmann’s parallelism plays key role in the first order asymptotic theory of quantum statistical estimation, and that the parallelism is closely related to Nagaoka’s quantum information geometry, which is already successfully applied to quantum estimation theory. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This post is the author's doctoral dissertation back in 1997. The dissertation covers following four kinds of problems: First, it studies achievable Cramer-Rao type bounds of various multi-parameter pure state models. Second, it relates CR-bouds, both of mixed state and pure states, with Berry-Uhlmann curvature. Roughly, it characterize how the statistical model differs from classical probability distribution family. Though the relation is rather qualitative for mixed states, the quantitative relation is obtained for pure state models. Third topic is relation between Berry-Uhlmann geometry and Amari-Nagaoka's quantum information geometry. Forth, various problems in quantum physics, uncertainty relations, measurement of temperature, time reversal symmetry, etc., are discussed using estimation theory.
Local asymptotic normality is a cornerstone of classical asymptotic statistics. Extending it to quantum statistical models entails formidable conceptual and technical challenges arising from their noncommutative nature, above all, the definition of a quantum likelihood ratio, whose very choice dictates the subsequent theoretical development. Employing quantum information geometry based on the symmetric logarithmic derivative, we formulate a quantum Lebesgue decomposition and the square-root likelihood ratio. These constructions provide a natural framework for a quantum analogue of Le Cam’s third lemma, the formulation of quantum local asymptotic normality, and the asymptotic representation results developed in our earlier work linking regular quantum statistical models to quantum Gaussian shift models. In this way, quantum information geometry emerges as a natural and unifying foundation for asymptotic quantum statistics.
Time reversal and duality of Markov processes will play a key role in the stochastic theory of quantum physics. We first discuss time reversal of time-homogeneous Markov processes. To handle the time reversal and (space-time) duality of time-inhomogeneous Markov processes we will define the so-called space-time Markov processes, and prove the main theorems of time reversal and duality. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We herein establish an asymptotic representation theorem for locally asymptotically normal quantum statistical models. This theorem enables us to study the asymptotic efficiency of quantum estimators, such as quantum regular estimators and quantum minimax estimators, leading to a universal tight lower bound beyond the i.i.d. assumption. This formulation complements the theory of quantum contiguity developed in the previous paper [Fujiwara and Yamagata, Bernoulli 26 (2020) 2105–2141], providing a solid foundation of the theory of weak quantum local asymptotic normality.
The hidden E 7 ( E 6 ) structure has been conjectured for the minimal model M4.5(M6.7){{\mathcal{M}}_{{4.5}}}({{\mathcal{M}}_{{6.7}}})M4.5​(M6.7​) perturbed by Φ 1.2 in the context of conformal field theory (CFT). Motivated by this, we examine the dilute A 4.6 models, which are expected to be corresponding lattice models. Thermodynamics of the equivalent one-dimensional quantum systems is analyzed via the quantum transfer matrix approach. Appropriate auxiliary functions, related to kinks in the theory, play a role in constructing functional relations among transfer matrices. We successfully recover the universal Y -systems and thereby the thermodynamic Bethe ansatz equations for E 6.7 from the dilute A 6.4 model, respectively. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.