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arXiv:2601.01858

A Survey of Bargmann Invariants: Geometric Foundations and Applications

Lin Zhang, Bing Xie

quant-phmath-phmath.FASubmitted Jan 5, 2026• Announced Jan 6, 2026• 0 citations
v1 · Jan 5, 2026v2 · Aug 2, 2026
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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.

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47 pages, 4 figures

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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.
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The geometrical arrangement of a set of quantum states can be completely characterized using relational information only. This information is encoded in the pairwise state overlaps, as well as in Bargmann invariants of higher degree written as traces of products of density matrices. We describe how to measure Bargmann invariants using suitable generalizations of the SWAP test. This allows for a complete and robust characterization of the projective-unitary invariant properties of any set of pure or mixed states. As applications, we describe basis-independent tests for linear independence, coherence, and imaginarity. We also show that Bargmann invariants can be used to characterize multi-photon indistinguishability.
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In this paper, we study the local unitary equivalence of quantum states on Cn⊗Cn\mathbb{C}^n\otimes\mathbb{C}^nCn⊗Cn, which is an important notion in quantum information theory. For the case n=2n=2n=2, the local unitary orbits of two-qubit states are completely determined by their local unitary Bargmann invariants. We show that the local unitary Bargmann invariants do not form a complete set of invariants of local unitary orbits for n⩾3n\geqslant 3n⩾3, which negatively answers a problem proposed by L. Zhang and B. Xie.
Abstract The geometrical arrangement of a set of quantum states can be completely characterized using relational information only. This information is encoded in the pairwise state overlaps, as well as in Bargmann invariants of higher degree written as traces of products of density matrices. We describe how to measure Bargmann invariants using suitable generalizations of the SWAP test. This allows for a complete and robust characterization of the projective-unitary invariant properties of any set of pure or mixed states. As applications, we describe basis-independent tests for linear independence, coherence, and imaginarity. We also show that Bargmann invariants can be used to characterize multi-photon indistinguishability.