Coexistency on Hilbert space effect algebras and a characterisation of its symmetry transformations

TL;DR

This paper investigates the coexistence relation among effects in Hilbert space effect algebras, characterizes symmetry transformations preserving this relation, and extends existing theorems in quantum measurement theory.

Contribution

It provides a characterization of effects with identical coexistence relations and describes all automorphisms of the effect algebra related to coexistence, expanding understanding of quantum measurement symmetries.

Findings

Identifies effects with the same coexistence properties.

Characterizes all automorphisms preserving coexistence.

Strengthens a theorem of Molnar.

Abstract

The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig's formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper's first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig's theorem. As a byproduct of our methods we also strengthen a theorem of Molnar.