Asymptotic quantum statistical estimation

Yu Watanabe

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.

Masahito Hayashi, Keiji Matsumoto

We derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus. It is also proven that this bound can be achieved in any statistical model in the qubit system. Moreover, we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models. That is, in such a statistical model, the quantum correlation can improve the accuracy of the estimation in an asymptotic setting.

Akio Fujiwara

Jonas Kahn

This thesis is entitled Quantum Local Asymptotic Normality and other questions of Quantum Statistics ,. Quantum statistics are statistics on quantum objects. In classical statistics, we usually start from the data. Indeed, if we want to predict the weather, and can measure the wind or the temperature, we can measure both. On the other hand the laws of physics themselves forbid us to measure simultaneously the speed and the position of an electron. We therefore have to start with the observed object itself, and choose the best measurement for our purposes.\n\nMy main result is that, for all statistical purposes, numerous copies of the same spin (magnetic state of an electron) is equivalent to a Gaussian state of a quantum harmonic oscillator, typically laser light. We can extend this to higher dimensions. As an application, we get an (asymptotically) optimal estimation scheme for unknown spins. The idea is to transform the spins into laser light, and use the already known optimal estimation methods for laser light.\n\nThe thesis furthermore includes four smaller problems, notably how to estimate a unitary (natural) evolution very quickly, and how best to decide which is the state of a quantum object, among a finite number

Sanjoy K. Mitter, Stephen K. Young