Quantum Phase via Geometry

Hoshang Heydari

Angelo C. M. Carollo, Davide Valenti, Bernardo Spagnolo

J H Hannay

Jincheng Zhang

The field of quantum geometry is experiencing a resurgence, driven by the desire to understand and manipulate the fundamental structure of quantum systems. This paper introduces the Phase-Field Quantum Geometry (PFQ) framework, a novel approach to representing and manipulating geometric configurations within quantum systems. PFQ leverages the concept of phase fields to dynamically construct complex geometric structures, offering a powerful tool for studying quantum phenomena and potentially enabling new quantum technologies. The core mechanism focuses on an automated "Phase Field Mapping" algorithm, which constructs intricate geometric representations from quantum data. This paper presents a detailed explanation of the PFQ framework, including its mathematical foundations, algorithm implementation, and potential applications across diverse quantum systems. The research explores the advantages of PFQ compared to existing geometric techniques and highlights its potential to unlock deeper insights into quantum mechanics.

Jean-Marc Lévy-Leblond

Chen Zeqian, Sun Mu, Zhi Yin

For giving a geometric interpretation of the observable-geometric phase with application to quantum computation, operator geometry on a finite-dimensional complex Hilbert space was developed by Chen (2020), i.e., geometry over the operator-principal G-bundle of the unitary operator group based on the observable space. In this paper, we further develop operator geometry on an infinite-dimensional separable complex Hilbert space. To this end, we use the theory of unbounded operators for describing the differential structure of infinite-dimensional observable spaces and defining quantum connections and quantum parallel transportation. Then, we introduce the notion of the observable-geometric phase for infinite-dimensional quantum systems and give its geometric interpretation in terms of operator geometry. Finally, we present a topological index for Floquet topological phases and prove a local formula for it in terms of the observable-geometric phases, which are measurable in the physical sense.