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Open Systems & Information Dynamics

Large Deviation Strategy for Inverse Problem II

Izumi Ojima, Kazuya Okamura

Published Sep 1, 2012• 2 citations
Publisher

Abstract

In the earlier paper [1], we have proposed the large deviation strategy (LDS) and discussed its first level. By efficient use of the central measure, we will establish a quantum version of Sanov's theorem, the Bayesian escort predictive state and the widely applicable information criteria for quantum states in LDS second level. Finally, these results are re-examined in the context of quantum estimation theory, and organized as quantum model selection, i.e., a quantum version of model selection.

DOI
10.1142/s1230161212500229

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In the earlier paper [1], we have proposed the large deviation strategy (LDS) and discussed its first level. By efficient use of the central measure, we will establish a quantum version of Sanov's theorem, the Bayesian escort predictive state and the widely applicable information criteria for quantum states in LDS second level. Finally, these results are re-examined in the context of quantum estimation theory, and organized as quantum model selection, i.e., a quantum version of model selection.
Taken traditionally as a no-go theorem against the theorization of inductive processes, Duhem-Quine thesis may interfere with the essence of statistical inference. This difficulty can be resolved by Micro-Macro duality \cite{Oj03, Oj05} which clarifies the importance of specifying the pertinent aspects and accuracy relevant to concrete contexts of scientific discussions and which ensures the matching between what to be described and what to describe in the form of the validity of duality relations. This consolidates the foundations of the inverse problem, induction method, and statistical inference crucial for the sound relations between theory and experiments. To achieve the purpose, we propose here Large Deviation Strategy (LDS for short) on the basis of Micro-Macro duality, quadrality scheme, and large deviation principle. According to the quadrality scheme emphasizing the basic roles played by the dynamics, algebra of observables together with its representations and universal notion of classifying space, LDS consists of four levels and we discuss its first and second levels in detail, aiming at establishing statistical inference concerning observables and states. By efficient use of the central measure, we will establish a quantum version of Sanov's theorem, the Bayesian escort predictive state and the widely applicable information criteria for quantum states in LDS second level. Finally, these results are reexamined in the context of quantum estimation theory, and organized as quantum model selection, i.e., a quantum version of model selection.
Abstract Quantum state estimation (or state tomography) is an indispensable task in quantum information processing. Because full state tomography that determines all elements of the density matrix is computationally demanding, one usually takes the strategy of assuming a certain model of quantum states and identifying the model parameters. However, it is difficult to make a valid assumption given little prior knowledge on a quantum state of interest, and thus we need a reasonable model selection method for quantum state estimation. Actually, in the classical statistical estimation theory, several types of information criteria have been established and widely used in practice for appropriately choosing a classical statistical model. In this study, we propose quantum information criteria for evaluating the quality of the estimated quantum state in terms of the quantum relative entropy, which is a natural quantum analogue of the classical information criterion defined in terms of Kullback–Leibler divergence. In particular, we derive two quantum information criteria depending on the type of an estimator for the quantum relative entropy; one uses the log-likelihood and the other uses the classical shadow. The general role of information criteria is to predict the performance of an estimated model for unseen data, although it is a function of only sampled data; this generalization capability of the proposed quantum information criteria is evaluated in numerical simulations.
In this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multi-site empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level-1 (empirical mean) rate function is independent of a model parameter while the level-2 (empirical measure) rate is not.