Jennifer Zhu, University of Waterloo
Categorical Limits of Quantum Graphs and Possibilities Induced by QuantumPseudometrics
Quantum graphs and quantum pseudometrics as defined by Kuperberg and Weaver have roots in quantum errorcorrection but have since been developed as subjects in their own right. The motivation for the first half of thisthesis is to build an infinite quantum graph from finite quantum graphs. The latter have been subjected to fargreater scrutiny due to their connections to categorical quantum theory, while the former have been somewhatneglected. To be precise, we define and take the categorical (co)limit of quantum graphs by developing a newnotion of morphism compatible with previous notions but carrying less baggage. The inspiration for the secondhalf follows from the (unpublished) theorem that pure states on a von Neumann algebra \mathcal{M} are givenby maximal filters in the projection lattice of \mathcal{M}. Upon the observation that points in a metric space$(X, d)$ with topology $\tau$ are also given by maximal filters $\tau$ and that quantum pseudometrics provide anotion of distance between projections in $B(\ell^2) \overline{\otimes} \mathcal M$, we are led to a notion ofdistance $f$ between pure states induced by these quantum pseudometrics. Also this function $f$ does not satisfythe triangle inequality, we make some parallels between it and David Lewis's conception of ``possible worlds.''
MC 2009