Scott Continuity in Generalized Probabilistic Theories

@inproceedings{Furber2019ScottCI,
  title={Scott Continuity in Generalized Probabilistic Theories},
  author={Robert Furber},
  booktitle={QPL},
  year={2019}
}
Scott continuity is a concept from domain theory that had an unexpected previous life in the theory of von Neumann algebras. Scott-continuous states are known as normal states, and normal states are exactly the states coming from density matrices. Given this, and the usefulness of Scott continuity in domain theory, it is natural to ask whether this carries over to generalized probabilistic theories. We show that the answer is no - there are infinite-dimensional convex sets for which the set of… 

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References

SHOWING 1-10 OF 35 REFERENCES

Categorical Equivalences from State-Effect Adjunctions

The surprising result that the equivalent subcategories consist of reflexive order-unit spaces and reflexive base-norm spaces, respectively, are the convex sets that can occur as state spaces in generalized probabilistic theories satisfying both the no-restriction hypothesis and its dual.

Domain theory

bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the

Information processing in generalized probabilistic theories

A framework in which a variety of probabilistic theories can be defined, including classical and quantum theories, and many others, is introduced, and a tensor product rule for combining separate systems can be derived.

Convex structures and operational quantum mechanics

A general mathematical framework called a convex structure is introduced. This framework generalizes the usual concept of a convex set in a real linear space. A metric is constructed on a convex

On an Algebraic generalization of the quantum mechanical formalism

One of us has shown that the statistical properties of the measurements of a quantum mechanical system assume their simplest form when expressed in terms of a certain hypercomplex algebra which is

The expectation monad in quantum foundations

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures, and leads to a new re-formulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.

Convexity, Duality and Effects

This paper describes some basic relationships between mathematical structures that are relevant in quantum logic and probability, namely convex sets, effect algebras, and a new class of functors that

Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure

General relativity is a deterministic theory with non-fixed causal structure. Quantum theory is a probabilistic theory with fixed causal structure. In this paper, we build a framework for

Handbook of Categorical Algebra

The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of

Topological Vector Spaces

This chapter presents the most basic results on topological vector spaces. With the exception of the last section, the scalar field over which vector spaces are defined can be an arbitrary,