Quantum Information on Spectral Sets

For convex optimization problems Bregman divergences appear as regret functions. Such regret functions can be defined on any convex set but if a sufficiency condition is added the regret function must be proportional to information divergence and the convex set must be spectral. Spectral set are sets where different orthogonal decompositions of a state into pure states have unique mixing coefficients. Only on such spectral sets it is possible to define well behaved information theoretic quantities like entropy and divergence. It is only possible to perform measurements in a reversible way if the state space is spectral. The most important spectral sets can be represented as positive elements of Jordan algebras with trace 1. This means that Jordan algebras provide a natural framework for studying quantum information. We compare information theory on Hilbert spaces with information theory in more general Jordan algebras, and conclude that much of the formalism is unchanged but also identify some important differences.Comment: 13 pages, 2 figures. arXiv admin note: text overlap with arXiv:1701.0101

Towards a Convenient Category of Topological Domains

We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models

A Convenient Category of Domains

We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual $omega$-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties

Action semantics in retrospect

This paper is a themed account of the action semantics project, which Peter Mosses has led since the 1980s. It explains his motivations for developing action semantics, the inspirations behind its design, and the foundations of action semantics based on unified algebras. It goes on to outline some applications of action semantics to describe real programming languages, and some efforts to implement programming languages using action semantics directed compiler generation. It concludes by outlining more recent developments and reflecting on the success of the action semantics project

Coloring Complexes and Combinatorial Hopf Monoids

We generalize the notion of coloring complex of a graph to linearized combinatorial Hopf monoids. These are a generalization of the notion of coloring complex of a graph. We determine when a combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the quasisymmetric function invariants associated to the combinatorial Hopf monoid. We show that the collection of all such coloring complexes forms a combinatorial Hopf monoid, which is the terminal object in the category of combinatorial Hopf monoids with convex characters. We also study several examples of combinatorial Hopf monoids.Comment: 37 pages, 5 figure

Multigraded Hilbert Series of noncommutative modules

In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the sum of the Hilbert series is a rational function. Moreover, a characterization of finite-dimensional algebras is obtained in terms of the nilpotency of a key matrix involved in the computations. Using this result, efficient variants of the methods are also developed for the computation of Hilbert series of truncated infinite-dimensional algebras whose (non-truncated) Hilbert series may not be rational functions. We consider some applications of the computation of multigraded Hilbert series to algebras that are invariant under the action of the general linear group. In fact, in this case such series are symmetric functions which can be decomposed in terms of Schur functions. Finally, we present an efficient and complete implementation of (standard) graded and multigraded Hilbert series that has been developed in the kernel of the computer algebra system Singular. A large set of tests provides a comprehensive experimentation for the proposed algorithms and their implementations.Comment: 28 pages, to appear in Journal of Algebr

Information is not about measurability

We present a simple example where the use of σ-algebras as a model of information leads to a paradoxical conclusion: a decisionmaker prefers less information to more. We then explain that the problem arises because the use of σ-algebras as the informational content of a signal is inadequate. We provide a characterization of the different models of information in the literature in terms of Blackwell’s theorem