On the homogeneity of the quantum transition probability

TL;DR

This paper explores the mathematical structure of quantum transition probabilities, revealing their maximum homogeneity in simple Euclidean Jordan algebras and characterizing state space boundaries topologically.

Contribution

It connects the homogeneity of quantum transition probabilities to Euclidean Jordan algebras and characterizes atomic state spaces through topology, offering new insights into quantum logic structures.

Findings

Transition probability is maximally homogeneous in all simple Euclidean Jordan algebras.

Atomic parts of these algebras can be characterized topologically.

Non-homogeneous cases occur with the E6-bioctonionic projective plane.

Abstract

In the years 1952 and 1965, H.-C. Wang and U. Hirzebruch showed that the two-point homogeneous compact spaces with convex metrics are isometric to the spheres, the real, complex, octonion projective spaces and the Moufang plane and as well to the sets of the minimal idempotents or pure states in the simple Euclidean Jordan algebras. Here we reveal the physical meaning of these mathematical achievements for the quantum mechanical transition probability. We show that this transition probability features a maximum degree of homogeneity in all simple Euclidean Jordan algebras, which includes common finite-dimensional Hilbert space quantum theory. The atomic parts of these algebras or, equivalently, the extreme boundaries of their state spaces can be characterized by purely topological means. This is an important difference to many other recent approaches that aim to distinguish the entire state spaces among the convex compact sets. An interesting case with non-homogeneous transition probability arises, when the $E_6$-symmetric bioctonionic projective plane is used as quantum logic.