Transition Probability (Fidelity) and Its Relatives

TL;DR

This paper explores the concept of transition probability (fidelity) in quantum systems, introducing amplitudes and a gauge theory framework linked to the Riemann-Bures metric for a deeper understanding.

Contribution

It presents a novel approach to quantum fidelity using amplitudes and establishes a gauge theory perspective connected to the Riemann-Bures metric.

Findings

Amplitudes provide an intuitive treatment of transition probabilities.

A gauge theory governing natural parallel transport is formulated.

Connections to the Riemann-Bures metric are elucidated.

Abstract

Transition Probability (fidelity) for pairs of density operators can be defined as "functor" in the hierarchy of "all" quantum systems and also within any quantum system. The introduction of "amplitudes" for density operators allows for a more intuitive treatment of these quantities, also pointing to a natural parallel transport. The latter is governed by a remarkable gauge theory with strong relations to the Riemann-Bures metric.