Categorical Limits of Quantum Graphs and Possibilities Induced by Quantum Pseudometrics

Abstract

Two monographs [Wea12] and [KW12] introduced new notions of quantum relations and quantum pseudometric spaces incorporating inspiration and techniques from a broad array of fields related to quantum theory.

We begin by investigating quantum relations; namely, we find a new formulation of a morphism of quantum relations. Under the general principle that classical functions should be dualized to contravariant maps between associated algebras in quantum theory, we use some operator space theory to analogously dualize the complement of a subset of vertices. This framework yields a representation independent expression of a morphism of quantum relations that aligns with previously representation dependent ones under the appropriate assumptions. Under these morphisms, the categorical (co)limit of a subclass of quantum relations has an obvious candidate. We also define these morphisms on the level of bimodules

We next investigate quantum pseudometrics with an emphasis on the quantum mechanical interpretation of the background and results. The motivational theorem (due to unpublished work by Farah and Weaver) is that pure states of a von Neumann algebra 𝓜 are in bijection with maximal filters in the projection lattice of 𝓜. Under the observation that the neighborhood filter of a point in a topological space is also a maximal filter and armed with a notion of distance 𝜌 between projections given by quantum pseudometrics, we investigate whether 𝜌 induces a notion of distance between pure states.