# Coexistence on Reflecting Hyperplane in Generalized Probability Theories

## TL;DR

This paper explores the structure of effect coexistence in generalized probability theories, focusing on reflecting hyperplanes in even-sided regular polygon state spaces, revealing conditions for unbiased effects and connecting to qubit effects.

## Contribution

It introduces the concept of reflecting hyperplanes in generalized probability theories and provides a necessary and sufficient condition for unbiased effects in even-sided polygon models.

## Key findings

- Reflecting hyperplanes contain all nontrivial extremal effects.
- Conditions for coexistence of unbiased effects are established.
- Results generalize known qubit effect properties.

## Abstract

The coexistence of effects in a certain class of generalized probability theories is investigated. The effect space corresponding to an even-sided regular polygon state space has a central hyperplane that contains all the nontrivial extremal effects. The existence of such a hyperplane, called a reflecting hyperplane, is tightly related to the point symmetry of the corresponding state space. The effects on such a hyperplane can be regarded as the (generalized) unbiased effects. A necessary and sufficient condition for a pair of unbiased effects in the even-sided regular polygon theories is presented. This result reproduces a low-dimensional analogue of known results of qubit effects in a certain limit.