6 research outputs found
On the Axiomatizability of Quantitative Algebras
Quantitative algebras (QAs) are algebras over metric spaces defined by
quantitative equational theories as introduced by the same authors in a related
paper presented at LICS 2016. These algebras provide the mathematical
foundation for metric semantics of probabilistic, stochastic and other
quantitative systems. This paper considers the issue of axiomatizability of
QAs. We investigate the entire spectrum of types of quantitative equations that
can be used to axiomatize theories: (i) simple quantitative equations; (ii)
Horn clauses with no more than equations between variables as hypotheses,
where is a cardinal and (iii) the most general case of Horn clauses. In
each case we characterize the class of QAs and prove variety/quasivariety
theorems that extend and generalize classical results from model theory for
algebras and first-order structures.Comment: 34 pages, this is an extended version of the paper with the same
title presented at LICS 201
Discrete equational theories
We introduce discrete equational theories where operations are induced by
those having discrete arities. We characterize the corresponding monads as
monads preserving surjections. Using it, we prove Birkhoff type theorems for
categories of algebras of discrete theories. This extends known results from
metric spaces to general symmetric monoidal closed categories.Comment: 13 page
Quantum channels as a categorical completion
We propose a categorical foundation for the connection between pure and mixed
states in quantum information and quantum computation. The foundation is based
on distributive monoidal categories.
First, we prove that the category of all quantum channels is a canonical
completion of the category of pure quantum operations (with ancilla
preparations). More precisely, we prove that the category of completely
positive trace-preserving maps between finite-dimensional C*-algebras is a
canonical completion of the category of finite-dimensional vector spaces and
isometries.
Second, we extend our result to give a foundation to the topological
relationships between quantum channels. We do this by generalizing our
categorical foundation to the topologically-enriched setting. In particular, we
show that the operator norm topology on quantum channels is the canonical
topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201
Enriched universal algebra
Following the classical approach of Birkhoff, we introduce enriched universal
algebra. Given a suitable base of enrichment , we define a language
to be a collection of -ary function symbols whose arities
are taken among the objects of . The class of -terms is
constructed recursively from the symbols of , the morphisms in
, and by incorporating the monoidal structure of .
Then, -structures and interpretations of terms are defined, leading
to enriched equational theories. In this framework we prove several fundamental
theorems of universal algebra, including the characterization of algebras for
finitary monads on as models of an equational theories, and
several Birkhoff-type theorems