Approximation in quantale-enriched categories

Our work is a fundamental study of the notion of approximation in V-categories and in (U,V)-categories, for a quantale V and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of V- and (U,V)-categories. We fully characterize continuous V-categories (resp. (U,V)-categories) among all cocomplete V-categories (resp. (U,V)-categories) in the same ways as continuous domains are characterized among all dcpos. By varying the choice of the quantale V and the notion of ideals, and by further allowing the ultrafilter monad to act on the quantale, we obtain a flexible theory of continuity that applies to partial orders and to metric and topological spaces. We demonstrate on examples that our theory unifies some major approaches to quantitative domain theory.Comment: 17 page

Contextual behavioural Metrics (Extended Version)

We introduce contextual behavioural metrics (CBMs) as a novel way of measuring the discrepancy in behaviour between processes, taking into account both quantitative aspects and contextual information. This way, process distances by construction take the environment into account: two (non-equivalent) processes may still exhibit very similar behaviour in some contexts, e.g., when certain actions are never performed. We first show how CBMs capture many well-known notions of equivalence and metric, including Larsen's environmental parametrized bisimulation. We then study compositional properties of CBMs with respect to some common process algebraic operators, namely prefixing, restriction, non-deterministic sum, parallel composition and replication.Comment: Extended version of a paper accepted for publication in proc. CONCUR 202

Partial metrizability in value quantales

[EN] Partial metrics are metrics except that the distance from a point to itself need not be 0. These are useful in modelling partially defined information, which often appears in computer science. We generalize this notion to study “partial metrics” whose values lie in a value quantale which may be other than the reals. Then each topology arises from such a generalized metric, and for each continuous poset, there is such a generalized metric whose topology is the Scott topology, and whose dual topology is the lower topology. These are both corollaries to our result that a bitopological space is pairwise completely regular if and only if there is such a generalized metric whose topology is the first topology, and whose dual topology is the second.This author wishes to acknowledge support for this research from the EPSRC of the United Kingdom (grant GR/S07117/01), and from the City University of New York (PSCCUNY grant 64472-00 33).Kopperman, RD.; Matthews, S.; Pajoohesh, H. (2004). Partial metrizability in value quantales. Applied General Topology. 5(1):115-127. https://doi.org/10.4995/agt.2004.2000SWORD1151275

Measurement spaces

The question of what should be meant by a measurement is tackled from a mathematical perspective whose physical interpretation is that a measurement is a process via which a finite amount of classical information is generated. This motivates a mathematical definition of space of measurements that consists of a topological stably Gelfand quantale whose open sets represent measurable physical properties. It also accounts for the distinction between quantum and classical measurements, and for the emergence of "classical observers." The latter have a relation to groupoid C*-algebras, and link naturally to Schwinger's notion of selective measurement

Resource Theories as Quantale Modules

We aim to counter the tendency for specialization in science by advancing a language that can facilitate the translation of ideas and methods between disparate contexts. The methods we address relate to questions of "resource-theoretic nature". In a resource theory, one identifies resources and allowed manipulations that can be used to transform them. Some of the main questions are: How to optimize resources? What are the trade-offs between them? Can a given resource be converted to another one via the allowed manipulations? Because of the ubiquity of such questions, methods for answering them in one context can be used to tackle corresponding questions in new contexts. The translation occurs in two stages. Firstly, concrete methods are generalized to the abstract language to find under what conditions they are applicable. Then, one can determine whether potentially novel contexts satisfy these conditions. Here, we mainly focus on the first part of this two-stage process. The thesis starts with a more thorough introduction to resource theories and our perspective on them in chapter 1. Chapter 2 then provides a selection of mathematical ideas that we make heavy use of in the rest of the manuscript. In chapter 3, we present two variants of the abstract framework, whose relations to existing ones are summarized in table 1.1. The first one, universally combinable resource theories, offers a structure in which resources, desired tasks, and resource manipulations may all be viewed as "generalized resources". Blurring these distinctions, whenever appropriate, is a simplification that lets us understand the abstract results in elementary terms. It offers a slightly distinct point of view on resource theories from the traditional one, in which resources and their manipulations are considered independently. In this sense, the second framework in terms of quantale modules follows the traditional conception. Using these, we make contributions towards the task of generalizing concrete methods in chapter 4 by studying the ways in which meaningful measures of resources may be constructed. One construction expresses a notion of cost (or yield) of a resource, summarized in its generalized form in theorems 4.21 and 4.22. Among other applications, this construction may be used to extend measures from a subset of resources to a larger domain—such as from states to channels and other processes. Another construction allows the translation of resource measures between resource theories. A particularly useful version thereof is the translation of measures of distinguishability to other resource theories, which we study in detail. Special cases include resource robustness and weight measures as well as relative entropy based measures quantifying minimal distinguishability from freely available resources. We instantiate some of these ideas in a resource theory of distinguishability in chapter 5. It describes the utility of systems with probabilistic behavior for the task of distinguishing between hypotheses, which said behavior may depend on

Resource Theories as Quantale Modules

We aim to counter the tendency for specialization in science by advancing a language that can facilitate the translation of ideas and methods between disparate contexts. The methods we address relate to questions of "resource-theoretic nature". In a resource theory, one identifies resources and allowed manipulations that can be used to transform them. Some of the main questions are: How to optimize resources? What are the trade-offs between them? Can a given resource be converted to another one via the allowed manipulations? Because of the ubiquity of such questions, methods for answering them in one context can be used to tackle corresponding questions in new contexts. The translation occurs in two stages. Firstly, concrete methods are generalized to the abstract language to find under what conditions they are applicable. Then, one can determine whether potentially novel contexts satisfy these conditions. Here, we mainly focus on the first part of this two-stage process. The thesis starts with a more thorough introduction to resource theories and our perspective on them in chapter 1. Chapter 2 then provides a selection of mathematical ideas that we make heavy use of in the rest of the manuscript. In chapter 3, we present two variants of the abstract framework, whose relations to existing ones are summarized in table 1.1. The first one, universally combinable resource theories, offers a structure in which resources, desired tasks, and resource manipulations may all be viewed as "generalized resources". Blurring these distinctions, whenever appropriate, is a simplification that lets us understand the abstract results in elementary terms. It offers a slightly distinct point of view on resource theories from the traditional one, in which resources and their manipulations are considered independently. In this sense, the second framework in terms of quantale modules follows the traditional conception. Using these, we make contributions towards the task of generalizing concrete methods in chapter 4 by studying the ways in which meaningful measures of resources may be constructed. One construction expresses a notion of cost (or yield) of a resource, summarized in its generalized form in theorems 4.21 and 4.22. Among other applications, this construction may be used to extend measures from a subset of resources to a larger domain—such as from states to channels and other processes. Another construction allows the translation of resource measures between resource theories. A particularly useful version thereof is the translation of measures of distinguishability to other resource theories, which we study in detail. Special cases include resource robustness and weight measures as well as relative entropy based measures quantifying minimal distinguishability from freely available resources. We instantiate some of these ideas in a resource theory of distinguishability in chapter 5. It describes the utility of systems with probabilistic behavior for the task of distinguishing between hypotheses, which said behavior may depend on