Statistical Modelling of Quantum Data

Abstract

The work presented in this thesis considers statistical models of quantum statistics. It sets up a framework to analyse experimental data, which is independent of any particular theory, being careful to eliminate any bias towards the theory of quantum mechanics. This framework allows us to separate the discussion of quantum statistics, which are the predictions of quantum theory, from the theory of quantum mechanics enabling an evaluation of alternative models in reproducing the same statistics. The framework allows one to evaluate the non-classicality of a process or phenomenon, without relying on their definitions in quantum mechanics. The two main non-classical phenomenon analysed in this thesis are contextuality and the weak value. We provide completely statistical definitions for both, which is completely independent of quantum mechanics. The two are defined as statistical properties of experimental data. The presence of contextuality in the data for a set of experiments is shown to pose certain obstructions to the statistical modelling of the data, specifically it is shown to require the state of the system to be updated in a non-Markovian way. A statistical model of this type is presented as proof of existence of statistical models for contextual data. The construction of this model is used to o_er insight into the mechanisms required to construct statistical models for contextual data. Additionally an explicit connection is established between contextuality and classical simulability for any sub-theory of quantum mechanics. This connection is used to evaluate the contextuality found in the qubit-stabilizer sub-theory to calculate a lower bound for the minimum number of classical bits a model would require to simulate the statistics. The weak value was previously thought of an a quintessentially quantum phenomenon. We analyse the dynamics of the process that leads to the weak value using classical Hamiltonian mechanics. A statistical analysis of the experiment explains the physical objects associated with the real and imaginary parts of the weak value and its emergence as a result of post-selection. The weak value calculated using classical Hamiltonian dynamics matches the quantum predictions.