## Ontological Models and Contextuality

##### Kerppo, Oskari (2017-10-17)

##### Tiivistelmä

The main question under investigation in this thesis was the reality of the quantum state. We approached this problem by studying a recent no-go result by Pusey, Barrett and Rudolph, namely the PBR theorem.

In Section 1 we provided a short introduction to quantum theory. A few relevant examples were covered, namely decompositions of quantum states, impossibility of the ignorance interpretation and the distinguishability of pure quantum states. Section 2 developed measure theoretic tools needed in the framework of ontological models. The Radon-Nikodym theorem was proven in order to justify the use of an alternative expression of the variational distance of probability measures.

In Section 3 we developed a general framework for ontological models. In short, the ontological models framework can be considered the most general framework currently available for hidden variable theories and no-go results. We used the measure theoretic tools of Section 2 to make the formalism rigorous.

Original research included in Section 3 was the characterization of a property called antidistinguishability for qubit states. A necessary and sufficient condition for antidistinguishability was given. It was then proven that any set of pure qubit states that is not antidistinguishable can be made antidistinguishable by adding a single carefully chosen state into the set.

In the end of Section 3, some original observations were presented. Namely, a specific connection with contextuality was proven for sometimes ψ-ontic and never ψ-ontic ontological models. However, it was then noted that these observations were not enough to exclude never ψ-ontic ontological models, a feat that would have provided a new no-go theorem on its own.

Contextuality in the ontological models framework was studied in Section 4. The main results were establishing preparation contextuality from the failure of maximal ψ-epistemicity and a proof of Bell's theorem based on preparation contextuality established this way. We concluded that ψ-ontic models cannot be Bell local based on these results. Finally in Section 5 the implications of ψ-ontology were discussed. By careful analysis of the results, it was noted that accepting the reality of the quantum state exposes explanatory gaps in the formalism. Possible solutions to the challenges introduced by these explanatory gaps were discussed through interpretations of quantum mechanics and the so-called measurement problem.

In Section 1 we provided a short introduction to quantum theory. A few relevant examples were covered, namely decompositions of quantum states, impossibility of the ignorance interpretation and the distinguishability of pure quantum states. Section 2 developed measure theoretic tools needed in the framework of ontological models. The Radon-Nikodym theorem was proven in order to justify the use of an alternative expression of the variational distance of probability measures.

In Section 3 we developed a general framework for ontological models. In short, the ontological models framework can be considered the most general framework currently available for hidden variable theories and no-go results. We used the measure theoretic tools of Section 2 to make the formalism rigorous.

Original research included in Section 3 was the characterization of a property called antidistinguishability for qubit states. A necessary and sufficient condition for antidistinguishability was given. It was then proven that any set of pure qubit states that is not antidistinguishable can be made antidistinguishable by adding a single carefully chosen state into the set.

In the end of Section 3, some original observations were presented. Namely, a specific connection with contextuality was proven for sometimes ψ-ontic and never ψ-ontic ontological models. However, it was then noted that these observations were not enough to exclude never ψ-ontic ontological models, a feat that would have provided a new no-go theorem on its own.

Contextuality in the ontological models framework was studied in Section 4. The main results were establishing preparation contextuality from the failure of maximal ψ-epistemicity and a proof of Bell's theorem based on preparation contextuality established this way. We concluded that ψ-ontic models cannot be Bell local based on these results. Finally in Section 5 the implications of ψ-ontology were discussed. By careful analysis of the results, it was noted that accepting the reality of the quantum state exposes explanatory gaps in the formalism. Possible solutions to the challenges introduced by these explanatory gaps were discussed through interpretations of quantum mechanics and the so-called measurement problem.