The Automorphism and Coexistence of The Quantum Effects

Abstract

This work is a study on the symmetry groups of quantum mechanics accompanied by its applications on the coexistence of qubit effects. We surveyed among some of the mappings known as preservers on the set of quantum effects for their role in conserving specific structures that are important in quantum mechanics. Since the preserves in this work mainly target the quantum effects, we provided principals of its corresponding algebra known as effect algebra. Then we advanced, to study the relations between some of these preservers (here we mean automorphisms) by channels operate on qubit effects. The channels are in convex form and we investigate the role of their convex parameters in conserving or breaking the coexistence of two qubit effects. The preserver that we are interested in is a bijective map which preserves the order and coexistence of effects in both directions. Of course, the coexistence of qubit effects can also be studied in the language of coexistence and compatibility of observables. Then our task is to study the coexistence of observables when there are only two effects in their outcome sets. This task has been studied for observables with arbitrary number of effects in their outcome sets [48]. The bounds they have found for the channels convex parameters are independent of that number. However, when the case narrows for limited number outcomes no analytic solution is available, and the study should be carried numerically. Here, we considered observables each having only two nontrivial effects in their out come sets, and asked for the demanding conditions that break their compatibility partially or completely.