Coexistence on reflecting hyperplane in generalized probability theories

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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 closely 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.

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