Domain Theory and Measurement

Abstract

Summary. Lecture notes on domain theory and measurement, driven by applications to physics, computer science and information theory, with a hint of provocation. Keye Martin

An Order-Theoretic Quantification of Contextuality

“…In this essay, I extend the work of Coecke and Martin [7,8] on states and measurements, within the neo-realist framework developed by Döring and Isham [9], to the concept of quantum contextuality. I begin with a review of the basic principles and definitions that will be used throughout this essay.…”

“…Together, they form a domain, (Σ, ). Given two objects ρ, σ ∈ Σ, the statement ρ σ is interpreted as saying that ρ contains (or carries) some information about σ, i.e., σ is "more informative" than ρ [8]. We take "information" to be anything that may be represented by a state ρ.…”

“…ρ n . The sense of decreasing partiality suggests that the information order is, in Martin's words, "going somewhere" [8]. That is:…”

“…ρ := {ρ, σ ∈ Σ : ρ σ} and ρ := {ρ, σ ∈ Σ : σ ρ} ↑ ρ := {ρ, σ ∈ Σ : ρ σ} and ↓ ρ := {ρ, σ ∈ Σ : σ ρ} (8) where the arrows suggest the "direction" of the information order. In fact, these may be viewed as defining a specific information order.…”

A Technique for Verifying Measurements

“…The measurement formalism expands the scope of domain theory [6,7] by providing new fixed point theorems, including those which apply to nonmonotonic functions, an informatic derivative for measuring the rate at which a process on a domain converges to a fixed point, methods for deriving distance from content, unified approaches to the continuous and discrete, a first order view of recursion ϕ = δ + ϕ • r which models iteration in its natural state and methods for relating algorithmic complexity to entropy via the structure of a domain.…”