## The Annals of Statistics

### Quantum local asymptotic normality based on a new quantum likelihood ratio

#### Abstract

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

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

Dates
First available in Project Euclid: 23 October 2013

https://projecteuclid.org/euclid.aos/1382547518

Digital Object Identifier
doi:10.1214/13-AOS1147

Mathematical Reviews number (MathSciNet)
MR3127863

Zentralblatt MATH identifier
1278.81050

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1382547518

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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.