A generic approach to the quantum mechanical transition probability

TL;DR

This paper generalizes the concept of quantum transition probability to orthomodular quantum logics, explores its connection to Jordan algebras, and presents a broad version of the no-cloning theorem with implications for quantum cryptography.

Contribution

It introduces a probabilistically motivated, general definition of quantum transition probability applicable to orthomodular quantum logics and links it to Jordan algebras, extending foundational quantum theory.

Findings

Generalized transition probability to orthomodular quantum logics

Established a broad quantum no-cloning theorem

Highlighted relationship between transition probability and Jordan algebras

Abstract

In quantum theory, the modulus-square of the inner product of two normalized Hilbert space elements is to be interpreted as the transition probability between the pure states represented by these elements. A probabilistically motivated and more general definition of this transition probability was introduced in a preceding paper and is extended here to a general type of quantum logics: the orthomodular partially ordered sets. A very general version of the quantum no-cloning theorem, creating promising new opportunities for quantum cryptography, is presented and an interesting relationship between the transition probability and Jordan algebras is highlighted.