Southwest Quantum Information and Technology

Joint measurement on the reflecting hyperplane in generalized probability theories

Presenting Author: Masatomo Kobayashi, Kyoto University
Contributing Author(s): Takayuki Miyadera

The existence of a pair of observables which is not jointly measurable is one of the most crucial aspects in quantum theory. The problem to find the necessary and sufficient conditions for effects to be coexistent is hard and has been only partially solved. It is, however, known that this peculiar property is not specific to the quantum theory in the general framework called Generalized Probability Theories. They have been studied from various points of view such as Bell's inequality, teleportaion, broadcasting and so on. In these articles, some authors indicated that the regular polygon systems are grouped into two series by the number of the vertexes, i.e. even or odd. We study the even-sided ones and show that the corresponding effect spaces have a nice symmetrical hyperplane which contains all (nontrivial) extremal effects and divides the whole effect space into　reflection symmetric two subsets. We call it "reflecting hyperplane". Analyzing the coexistence problem in the polygon systems, we give necessary and sufficient conditions for a pair of effects on the hyperplane to be coexistent. Furthermore, we examine general systems (other than regular polygons) which have the reflecting hyperplane and show that the volume of the set of all effects coexistent with a nontrivial extreme effect is vanishing.

(Session 5 : Thursday from 5:00pm - 7:00pm)