The Annals of Statistics

Quantum local asymptotic normality based on a new quantum likelihood ratio

Koichi Yamagata, Akio Fujiwara, and Richard D. Gill

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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.

Article information

Ann. Statist., Volume 41, Number 4 (2013), 2197-2217.

First available in Project Euclid: 23 October 2013

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Zentralblatt MATH identifier

Primary: 81P50: Quantum state estimation, approximate cloning
Secondary: 62F12: Asymptotic properties of estimators

Quantum local asymptotic normality Holevo bound quantum log-likelihood ratio


Yamagata, Koichi; Fujiwara, Akio; Gill, Richard D. Quantum local asymptotic normality based on a new quantum likelihood ratio. Ann. Statist. 41 (2013), no. 4, 2197--2217. doi:10.1214/13-AOS1147.

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Supplemental materials

  • Supplementary material: Supplementary material to “Quantum local asymptotic normality based on a new quantum likelihood ratio”. Section A is devoted to proofs of Lemma 2.6, Theorems 2.9 and 2.10, Corollary 2.11, and Theorem 3.1. Section B is devoted to a brief account of quantum estimation theory, including quantum logarithmic derivatives, the commutation operator, the Holevo bound, estimation theory for quantum Gaussian shift models and for pure state models.