15,643 research outputs found
Representation of States on Effect-Tribes and Effect Algebras by Integrals
We describe -additive states on effect-tribes by integrals.
Effect-tribes are monotone -complete effect algebras of functions where
operations are defined by points. Then we show that every state on an effect
algebra is an integral through a Borel regular probability measure. Finally, we
show that every -convex combination of extremal states on a monotone
-complete effect algebra is a Jauch-Piron state.Comment: 20 page
Sharp and Unsharp Quantum Effects
AbstractA survey of the algebraic and the statistical properties of sharp and unsharp quantum effects is presented. We begin with a discussion and a comparison of four types of probability theories, the sharp and unsharp classical and quantum theories. A structure called an effect algebra that generalizes and unifies all four of these probability theories, is then considered. Finally, we present some recent investigations on tensor products and quotients of effect algebras. Examples and representative results for the various theories are discussed
A Categorical Construction of the Real Unit Interval
The real unit interval is the fundamental building block for many branches of
mathematics like probability theory, measure theory, convex sets and homotopy
theory. However, a priori the unit interval could be considered an arbitrary
choice and one can wonder if there is some more canonical way in which the unit
interval can be constructed. In this paper we find such a construction by using
the theory of effect algebras. We show that the real unit interval is the
unique non-initial, non-final irreducible algebra of a particular monad on the
category of bounded posets. The algebras of this monad carry an order,
multiplication, addition and complement, and as such model much of the
operations we need to do on probabilities. On a technical level, we show that
both the categories of omega-complete effect algebras as well as that of
omega-complete effect monoids are monadic over the category of bounded posets
using Beck's monadicity theorem. The characterisation of the real unit interval
then follows easily using a recent representation theorem for omega-complete
effect monoids.Comment: 13 pages + 2 page appendi
States on pseudo effect algebras and integrals
We show that every state on an interval pseudo effect algebra satisfying
some kind of the Riesz Decomposition Properties (RDP) is an integral through a
regular Borel probability measure defined on the Borel -algebra of a
Choquet simplex . In particular, if satisfies the strongest type of
(RDP), the representing Borel probability measure can be uniquely chosen to
have its support in the set of the extreme points of $K.
The Lattice and Simplex Structure of States on Pseudo Effect Algebras
We study states, measures, and signed measures on pseudo effect algebras with
some kind of the Riesz Decomposition Property, (RDP). We show that the set of
all Jordan signed measures is always an Abelian Dedekind complete -group.
Therefore, the state space of the pseudo effect algebra with (RDP) is either
empty or a nonempty Choquet simplex or even a Bauer simplex. This will allow
represent states on pseudo effect algebras by standard integrals
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