A Survey of Bargmann Invariants: Geometric Foundations and Applications

TL;DR

This survey comprehensively reviews Bargmann invariants, emphasizing their geometric significance in quantum mechanics and their applications in quantum information, including state characterization and entanglement detection.

Contribution

It provides a unified overview of Bargmann invariants' geometric foundations and demonstrates their practical applications in quantum state analysis and entanglement detection.

Findings

Bargmann invariants characterize the geometry of quantum state space.

They enable the classification of quantum states under local unitary transformations.

Applications include entanglement detection without full state tomography.

Abstract

Bargmann invariants, a class of gauge-invariant quantities arising from the overlaps of quantum state vectors, provide a profound and unifying framework for understanding the geometric structure of quantum mechanics. This survey offers a comprehensive overview of Bargmann invariants, with a particular focus on their role in shaping the informational geometry of the state space. The core of this review demonstrates how these invariants serve as a powerful tool for characterizing the intrinsic geometry of the space of quantum states, leading to applications in determining local unitary equivalence and constructing a complete set of polynomial invariants for mixed states. Furthermore, we explore their pivotal role in modern quantum information science, specifically in developing operational methods for entanglement detection without the need for full state tomography. By synthesizing historical context with recent advances, this survey aims to highlight Bargmann invariants not merely as mathematical curiosities, but as essential instruments for probing the relational and geometric features of quantum systems.