Local asymptotic normality in quantum statistics

Local asymptotic normality is a key concept in mathematical statistics, which underlies the asymptotic achievability of the Cramer-Rao bound and the asymptotic normality of optimal estimators.

In this presentation, I will review some of the current progress in extending this concept to the field of quantum statistics. In quantum estimation, one is given “quantum data” in the form of a quantum system prepared in a state that depends on an unknown parameter, and would like to estimate the parameter based on “classical data” obtained by measuring the system. This raises the problem of finding optimal measurements which extract maximum amount of information about the parameter.

Quantum local asymptotic normality (QLAN) shows that in the limit of large “sample size”, certain quantum statistical models can be approximated by simpler Gaussian models where the unknown parameter is encoded linearly in the mean of the canonical variables. This simplification provides a procedures for constructing optimal estimators with normally distributed errors. I will discuss two settings in which QLAN holds: ensembles of identical, independent systems, and quantum Markov chains.

This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.