Coexistence of Hilbert space effects and orthogonality

TL;DR

This paper explores the coexistence and compatibility properties of effects in Hilbert spaces, introducing new concepts like partial ortho-coexistence and generalized compatibility, with geometric insights in two-dimensional cases.

Contribution

It introduces the notions of partial ortho-coexistence and generalized compatibility, expanding understanding of effect coexistence in Hilbert spaces.

Findings

Absolutely compatible effects are coexistent and partially orthogonal.

Introduction of partial ortho-coexistence concept.

Geometric behavior of generalized compatibility in 2x2 matrices.

Abstract

In this paper, we show that every pair of absolutely compatible Hilbert space effects are coexistent and exhibit a partial orthogonality property. We introduce the notion of partially ortho-coexistence. We generalize absolute compatibility to obtain more examples of partially ortho-coexistent pairs and introduce the notion of generalized compatibility. In the case of $\mathbb{M}_2$, we discuss a geometric behaviour of the generalized compatibility.