Exchangeability theorems as categorical limits of probabilistic and quantum processes

Thesis

The topic of this thesis is a sequence of results which re-situate exchangeability theorems from probability theory in categories of probabilistic and quantum processes. These theorems take sequences of events, say random variables, random matrices, or quantum states, and study what behaviours are possible after imposing invariance of the action of finite permutations of the elements of the sequence. These are often philosophically motivated, in the classical case by subjectivist foundations of probability, in the quantum case by the school of quantum Bayesianism. In chapters 3 and 4, quantum and classical de Finetti theorems are generalised to limits of diagrams in appropriate categories, moving from statements about measures and quantum states to theorems about parameterised random measures and quantum states. The latter also considers the use of multisets for encoding exchangeability. In chapter 5, an approach using coalgebras of functors as theories of systems is taken. Instead of looking at sequences of events, we consider processes, coalgebras, that take a parameter and probabilistically return both an output and an updated parameter. Such coalgebras can be exchangeable, and it is shown that these coalgebras have a universal object similar to in the limit objects of the previous chapters, namely a final exchangeable coalgebra. Both the multiset and coalgebraic theorems are given quantum analogues.

Authors: Summers, N

• DOI: 10.5287/ora-ermd4qrd8
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: category theory; De Finetti theorem; categorical quantum foundations; categorical limits; quantum de finetti theorem; categorical probability theory
• Subjects: Symmetry (Mathematics); Probability measures; Triples, Theory of; Compact spaces; Categories (Mathematics)