<!-- *** Custom HTML *** --> We study asymptotically optimal statistical inference concerning the unknown state of $N$ identical quantum systems, using two complementary approaches: a “poor man’s approach” based on the van Trees inequality, and a rather more sophisticated approach using the recently developed quantum form of LeCam’s theory of Local Asymptotic Normality.
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
Observations or measurements taken of a quantum system (a small number of fundamental particles) are inherently random. If the state of the system depends on unknown parameters, then the distribution of the outcome depends on these parameters too, and statistical inference problems result. Often one has a choice of what measurement to take, corresponding to different experimental set-ups or settings of measurement apparatus. This leads to a design problem--which measurement is best for a given statistical problem. This paper gives an introduction to this field in the most simple of settings, that of estimating the state of a spin-half particle given n independent copies of the particle. We show how in some cases asymptotically optimal measurements can be constructed. Other cases present interesting open problems, connected to the fact that for some models, quantum Fisher information is in some sense non-additive. In physical terms, we have non-locality without entanglement.
Observations or measurements taken of a quantum system (a small number of fundamental particles) are inherently random.If the state of the system depends on unknown parameters, then the distribution of the outcome depends on these parameters too, and statistical inference problems result.Often one has a choice of what measurement to take, corresponding to different experimental set-ups or settings of measurement apparatus.This leads to a design problem-which measurement is best for a given statistical problem.This paper gives an introduction to this field in the most simple of settings, that of estimating the state of a spin-half particle given n independent copies of the particle.We show how in some cases asymptotically optimal measurements can be constructed.Other cases present interesting open problems, connected to the fact that for some models, quantum Fisher information is in some sense non-additive.In physical terms, we have non-locality without entanglement.
Summary Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems. Furthermore, developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory. The present paper reviews this field and proposes and interrelates some new concepts for an extension of classical statistical inference to the quantum context.
Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems. Furthermore, theoretical developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory. The present paper reviews this field and proposes and interrelates a number of new concepts for an extension of classical statistical inference to the quantum context.