Contexts in Convex and Sequential Effect Algebras

Abstract

A convex sequential effect algebra (COSEA) is an algebraic system with three physically motivated operations, an orthogonal sum, a scalar product and a sequential product. The elements of a COSEA correspond to yes-no measurements and are called effects. In this work we stress the importance of contexts in a COSEA. A context is a finest sharp measurement and an effect will act differently according to the underlying context with which it is measured. Under a change of context, the possible values of an effect do not change but the way these values are obtained may be different. In this paper we discuss direct sums and the center of a COSEA. We also consider conditional probabilities and the spectra of effects. Finally, we characterize COSEA's that are isomorphic to COSEA's of positive operators on a complex Hilbert space. These result in the traditional quantum formalism. All of this work depends heavily on the concept of a context.

A Theory of Entanglement

“…A context given by an orthonormal basis A = {φ i } can be thought of as giving a partial view of a quantum system. In order to obtain a total view we must consider various contexts [11,12]. We say that A ∈ L(H) is measurable with respect to A if AP φ i = P φ i A for every i.…”

“…In this case, φ i is an eigenvector of A with eigenvalue φ i , Aφ i = E φ i (A). The only operators accurately described by A are the operators that are measurable with respect to A [11,12]. We define the context coefficient of A with respect to A by…”

On the properties of spectral effect algebras

“…In this section we are going to prove the main result that answers the open question from [13] about the possible number of contexts in a spectral effect algebra. Proposition 6.…”

Three characterisations of the sequential product

“…Note: Gudder also studied convex sequential effect algebras in [6]. There the additional condition that the space is spectral was required.…”

“…This is a set equipped with a partial addition representing fuzzy conjunction, a complement representing negation, and a binary operation representing the sequential product. A sequential effect algebra has been shown to exhibit some of the same properties present in the set of quantum effects [7,6].…”