arXiv:1703.07535

Noncommutative Lebesgue decomposition with application to quantum local asymptotic normality

Akio Fujiwara, Koichi Yamagata

Published Mar 22, 2017• 1 citation

Abstract

We develop a theory of local asymptotic normality in the quantum domain based on a noncommutative extension of the Lebesgue decomposition. This formulation gives a substantial generalization of the previous paper [Yamagata, Fujiwara, and Gill (2013). Ann. Statist., 41, 2197-2217.], extending the scope of the quantum local asymptotic normality to a wider class of quantum statistical models that comprise density operators of mixed ranks.

DOI: 10.48550/arxiv.1703.07535

Related papers

Noncommutative Lebesgue decomposition and contiguity with applications in quantum statistics

Akio Fujiwara, Koichi Yamagata

We herein develop a theory of contiguity in the quantum domain based upon a novel quantum analogue of the Lebesgue decomposition. The theory thus formulated is pertinent to the weak quantum local asymptotic normality introduced in the previous paper [Yamagata, Fujiwara, and Gill, Ann. Statist. 41 (2013) 2197–2217], yielding substantial enlargement of the scope of quantum statistics.

arXiv:1804.03510BernoulliApr 27, 20206 citations

Foundations of quantum local asymptotic normality via information geometry

Akio Fujiwara

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.

Information GeometryMay 19, 20260 citations

Efficiency of estimators for locally asymptotically normal quantum statistical models

Akio Fujiwara, Koichi Yamagata

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.

arXiv:2209.00832The Annals of StatisticsJun 1, 20234 citations

Quantum local asymptotic normality based on a new quantum likelihood ratio

Koichi Yamagata, Akio Fujiwara, Richard David Gill

We develop a theory of local asymptotic normality in the quantum domain based on a novel quantum analogue of the log-likelihood ratio. This formulation is applicable to any quantum statistical model satisfying a mild smoothness condition. As an application, we prove the asymptotic achievability of the Holevo bound for the local shift parameter.

arXiv:1210.3749The Annals of StatisticsAug 1, 2013117 citations

Unbounded Local Observables in Quantum Statistical Mechanics

Geoffrey L. Sewell

The algebraic formulation of quantum statistical mechanics is extended so as to include local unbounded observables. We start, in a usual way, with a C*-algebra U of quasilocal bounded observables in Fock space HF, an arbitrary, locally normal state φ on U, and a corresponding Gel'fand-Naimark-Segal (GNS) representation Rφ of U in a Hilbert space Hφ. We then construct a set QL of local, closed operators whose domains are dense in HF, such that QL includes all the local observables of the system. The representation Rφ is then extended so as to provide a *-homomorphism of HL into the closed, densely defined operators in Hφ. Correspondingly, a number of results previously established for the local bounded observables are extended to the unbounded ones. For appropriate classes of locally normal states, these extended results include the Kubo-Martin-Schwinger boundary conditions, the spatially asymptotic and ergodic properties of space-correlation functions, and the temporally ergodic properties of time-correlation functions. It is also shown that, for locally normal Gibbs states, the time correlations between elements of a specified subset of QL are thermodynamical limits of the corresponding correlations for finite systems.

Journal of Mathematical PhysicsJun 1, 197017 citations

Localization of quantum states within subspaces from the Lebesgue decomposition

L. L. Salcedo

Relying on the Lebesgue-type decomposition of positive operators, this work introduces a rigorous notion of localization probability of a quantum state within a given subspace of its Hilbert space. A non-negative operator A is uniquely decomposed as A=B+C, where B is the maximal positive operator supported inside the chosen subspace and C has support disjoint from it. The localized component B can be expressed via the Schur complement and characterized through an A-dependent inner product and suitable trace inequalities. For quantum states, this yields a probability lambda that a state rho be completely contained in a subspace, which is strictly more restrictive than the usual overlap probability Tr(P rho) and enjoys concavity and super-additivity properties. The resulting framework admits natural interpretations in quantum information, including entropic aspects and a potential cryptographic masking scheme based on the uniqueness of the decomposition.

quant-pharXiv:2601.09817Jan 16, 20260 citations