Local asymptotic normality in quantum statistics

Abstract

The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family varphi_{θ_{0}+ u/\sqrt{n}}^{n} consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state phi_{u} of an algebra of canonical commutation relations. The convergence holds for all "local parameters" u\in R^{m} such that theta=theta_{0}+ u/sqrt{n} parametrizes a neighborhood of a fixed point theta_{0}\in Theta\subset R^{m}. In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and investigate the relation between the two notions of convergence. For reader's convenience and completeness we review the relevant results of the classical as well as the quantum theory.

41 pages, to appear in Commun. Math. Phys

References in corpus (8)

• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• Efficient Quantum State Estimation by Continuous Weak Measurement and Dynamical Control
• Local asymptotic normality for qubit states
• Optimal estimation of qubit states with continuous time measurements
• Asymptotic performance of optimal state estimation in quantum two level system
• Optimal cloning of mixed Gaussian states
• Quantum estimation and the quantum central limit theorem
• Statistical Model with Measurement Degree of Freedom and Quantum Physics

Cited by in corpus (47)

• Multi-parameter estimation beyond Quantum Fisher Information
• True precision limits in quantum metrology
• Local asymptotic normality for finite dimensional quantum systems
• Quantum local asymptotic normality based on a new quantum likelihood ratio
• Measurement of damping and temperature: Precision bounds in Gaussian dissipative channels
• On Asymptotic Quantum Statistical Inference
• Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation
• Fundamental limits to frequency estimation: A comprehensive microscopic perspective
• Quantum metrology in the presence of limited data
• Precision Limits in Quantum Metrology with Open Quantum Systems
• Fisher information and asymptotic normality in system identification for quantum Markov chains
• Optimal probes and error-correction schemes in multi-parameter quantum metrology
• Incompatibility in Quantum Parameter Estimation
• Quantum learning: optimal classification of qubit states
• Multiple phase estimation for arbitrary pure states under white noise
• Optimal estimation of qubit states with continuous time measurements
• Quantum learning of coherent states
• Multiple phase estimation in quantum cloning machines
• Equivalence classes and local asymptotic normality in system identification for quantum Markov chains
• Fourier Analytic Approach to Phase Estimation
• Information geometry and local asymptotic normality for multi-parameter estimation of quantum Markov dynamics
• Precision bounds in noisy quantum metrology
• Asymptotic inference in system identification for the atom maser
• Quantum incompatibility of channels with general outcome operator algebras
• Quantum teleportation benchmarks for independent and identically-distributed spin states and displaced thermal states
• Fourier Analytic Approach to Quantum Estimation of Group Action
• Compression for quantum population coding
• Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements
• Asymptotically optimal purification and dilution of mixed qubit and Gaussian states
• Quantum hypothesis testing for quantum Gaussian states: Quantum analogues of chi-square, t and F tests
• Local asymptotic equivalence of pure quantum states ensembles and quantum Gaussian white noise
• Maximum likelihood versus likelihood-free quantum system identification in the atom maser
• Minimal sufficient statistical experiments on von Neumann algebras
• Invertible condition of quantum Fisher information matrix for a mixed qubit
• Attainability of the Holevo-Cramér-Rao bound for two-qubit 3D magnetometry
• Accessible information without disturbing partially known quantum states on a von Neumann algebra
• Efficiency of estimators for locally asymptotically normal quantum statistical models
• Directed-completeness of quantum statistical experiments in the randomization order
• Asymptotically optimal quantum channel reversal for qudit ensembles and multimode Gaussian states
• Quantum Minimax Theorem
• Loss of Memory and Convergence of Quantum Markov Processes
• Topics in Estimation of Quantum Channels
• Estimating quantum Markov chains using coherent absorber post-processing and pattern counting estimator
• Quantum non demolition measurements: parameter estimation for mixtures of multinomials
• Infinite dimensionality of the post-processing order of measurements on a general state space
• Compression of quantum shallow-circuit states
• Noncommutative Lebesgue decomposition and contiguity with applications in quantum statistics