Coexistence of Hilbert space effects and orthogonality

Johan Åberg

Timothy J. Moreland, Stanley Gudder

György Pál Gehér, Peter Šemrl

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 Molnár.

Roberto Giuntini, Antonio Ledda, Francesco Paoli

Michiya Mori, Peter Šemrl

Abstract Let H be a finite-dimensional Hilbert space, dim H ⩾ 2. We prove that every continuous coexistency preserving map on the effect algebra E ( H ) is either a standard automorphism of E ( H ), or a standard automorphism of E ( H ) composed with the orthocomplementation. We present examples showing the optimality of the result.

Michiya Mori, Peter Šemrl

Let $H$ be a finite-dimensional Hilbert space, $\dim H \ge 2$. We prove that every continuous coexistency preserving map on the effect algebra $E(H)$ is either a standard automorphism of $E(H)$, or a standard automorphism of $E(H)$ composed with the orthocomplementation. We present examples showing the optimality of the result.