Uncertainty relations in quantum theory

Abstract

Uncertainty relations are commonly praised as one of the central pillars of quantum theory. Usually, they are taught in the first weeks of a beginners lecture, and introduced in the ?rst chapters of a textbook. However, their precise operational meaning and a formulation in a general context, i.e. beyond the example of position and momentum observables, are often left out. The reasoning for this is twofold: On one hand, an exact operational definition of uncertainty, indeterminacy and a corresponding uncertainty principle has been the content of many debates since the early days of quantum mechanics until today. From a modern perspective, we have the consent that there are at least the two notions of preparation and measurement uncertainty: the first notion prohibits the existence of dissipation free states and the latter one the existence of error free joint measurements. On the other hand, we have that, the mathematical tools, which are needed for comprehensive treatment of uncertainty relations in a general context, are still under development and usually go far beyond the mathematical level of an introductory course. In this thesis we will investigate these two notions of uncertainty, their corre- sponding uncertainty relations, such as their interplay. The aim of this thesis is to give answers to the central questions: (1.) Which quantities should be used to formulate uncertainty? (2.) How can we compute uncertainty relations for those? (3.) Are there connections between the two notions of uncertainty? We will do this, whenever possible, in a most general context and with a focus on relevant examples, otherwise. Therefore, we will consider constructions of mea- surement errors and deviation measures that quantify uncertainty, based on, so called, cost functions. Commonly used uncertainty measures like variances, en- tropies, and the Hamming distance are examples for these. We will investigate the structure of the corresponding uncertainty relations and provide several methods that enable us to compute them. The third question is addressed by a theorem that shows, for sharp observables, that measurement uncertainty relations can be lower bounded by preparation uncertainty relations, whenever the same cost function is used.