On the optimization of nonlocality proofs inquantum statistics

Abstract

This master thesis is centered around the the mathematical framework of quantum probability and quantum statistics. Quantum statistics is the axiomatization of the physical theory of quantum mechanics pioneered by von Neumann in the 1930s. In contrast to many other physical theories which are described by deterministic models quantum mechanics is a stochastic theory. In fact, quantum probability can be viewed as an extension to Kolomogorov’s “classical” probability theory. Especially in the last decade, due to first physical realizations of quantum computational systems and the rise of quantum information theory, the subject of quantum statistics became more and more important and many applications for mathematical statisticians and probabilists opened up. In this thesis we will be concerned with the optimization of so-called nonlocality proofs which are methods to show the “non-classicality” of certain probability distributions within the framework of quantum statistics. In particular, one is interested in measuring the statistical strength of such nonlocality proofs, sometimes called Bell tests. One of the main results of this thesis is the analysis of certain Bell tests where the corresponding measure space is described by an infinite dimensional separable Hilbert space, corresponding to infinitely many possible outcomes. In particular, this gives numerical evidence for a new quantum Bell inequality describing the boundary of the space of quantum probability distributions for the considered setting. These results have been published in the following letter, S. Zohren and R. D. Gill, “Maximal violation of the Collins- Gisin-Linden-Massar-Popescu inequality for infinite dimensional states” submitted to Phys. Rev. Lett..