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arXiv:2412.09070 · Physical Review A

Geometry of sets of Bargmann invariants

Lin Zhang, Bing Xie, Bo Li

Published Apr 11, 2025• 7 citations
arXivHTMLPDF

Abstract

Certain unitary invariants, known as Bargmann invariants or multivariate traces of quantum states, have recently gained attention due to their applications in quantum information theory. However, determining the boundaries of sets of Bargmann invariants remains a theoretical challenge. In this study, we address the problem by developing a unified, dimension-independent formulation that characterizes the sets of the third and fourth Bargmann invariants. In particular, our result for the set of fourth Bargmann invariants confirms the conjecture given by Fernandes et al. [Phys. Rev. Lett. 133, 190201 (2024)]. Based on the obtained results, we conjecture that the unified, dimension-independent formulation of the boundaries for sets of third- and fourth-order Bargmann invariants may extend to the general case of the $n\mathrm{th}$-order Bargmann invariants. These results deepen our understanding of the fundamental physical limits within quantum mechanics and pave the way for novel applications of Bargmann invariants in quantum information processing and related fields.

DOI
10.1103/physreva.111.042417

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Certain unitary invariants, known as Bargmann invariants or multivariate traces of quantum states, have recently gained attention due to their applications in quantum information theory. However, determining the boundaries of sets of Bargmann invariants remains a theoretical challenge. In this study, we address the problem by developing a unified, dimension-independent formulation that characterizes the sets of the third and fourth Bargmann invariants. In particular, our result for the set of fourth Bargmann invariants confirms the conjecture given by Fernandes et al. [Phys. Rev. Lett. 133, 190201 (2024)]. Based on the obtained results, we conjecture that the unified, dimension-independent formulation of the boundaries for sets of third- and fourth-order Bargmann invariants may extend to the general case of the nthn\mathrm{th}nth-order Bargmann invariants. These results deepen our understanding of the fundamental physical limits within quantum mechanics and pave the way for novel applications of Bargmann invariants in quantum information processing and related fields.
The imaginary in quantum theory plays a crucial role in describing quantum coherence and is widely applied in quantum information tasks such as state discrimination, pseudorandomness generation, and quantum metrology. A recent paper by Fernandes et al. [C. Fernandes, R. Wagner, L. Novo, and E. F. Galvão, Phys. Rev. Lett. 133, 190201 (2024) ] showed how to use the Bargmann invariant to witness the imaginarity of a set of quantum states. In this work, we delve into the structure of Bargmann invariants and their quantum realization in qubit systems. First, we present a characterization of special sets of Bargmann invariants (also studied by Fernandes et al. for a set of four states) for a general set of nnn quantum states. Then, we study the properties of the relevant Bargmann invariant set Bn\mathcal{B}_nBn​ and its quantum realization in qubit systems. Our results provide new insights into the structure of Bargmann invariants, contributing to the advancement of quantum information techniques, particularly within qubit systems.
Bargmann invariants, a class of unitary-invariant quantities arising from the overlaps of quantum state vectors, provide a profound and unifying framework for understanding the relative geometry of the projective Hilbert space. This survey offers a comprehensive overview of their theoretical characterization and practical applications, with particular emphasis on recent progress in determining the full structure of their admissible set. 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. On the operational side, we review the cycle-test quantum circuit for the direct estimation of Bargmann invariants without full state tomography, and demonstrate their utility in witnessing quantum imaginary, discriminating local unitary equivalence, and detecting entanglement via partial-transpose moments---with explicit complete invariant sets provided for two-qubit systems. By connecting fundamental geometric classification with experimentally feasible estimation protocols,this survey establishes Bargmann invariants as indispensable probes of the relational, noncommutative, and geometric structure of quantum states, and identifies key open problems for multipartite high-dimensional systems and for quantum resource theories.
The imaginary in quantum theory plays a crucial role in describing quantum coherence and is widely applied in quantum information tasks such as state discrimination, pseudorandomness generation, and quantum metrology. A recent paper by Fernandes et al. [Phys. Rev. Lett. 133, 190201 (2024)] showed how to use the Bargmann invariant to witness the imaginarity of a set of quantum states. In this work, we delve into the structure of Bargmann invariants and their quantum realization in qubit systems. First, we present a characterization of special sets of Bargmann invariants (also studied by Fernandes et al. for a set of four states) for a general set of nnn quantum states. Then, we study the properties of the relevant Bargmann invariant set Bn{\mathcal{B}}_{n}Bn​ and its quantum realization in qubit systems. Our results provide new insights into the structure of Bargmann invariants, contributing to the advancement of quantum information techniques, particularly within qubit systems.
Considerable effort has been devoted to developing techniques for witnessing and characterizing quantum resources that emerge from collective properties of a set of states. In this context, Bargmann invariants play a central role: they witness coherence and related resources, and underpin important applications. In this work, we introduce a unified formalism that fully characterizes and organizes the capability of Bargmann invariants to witness different manifestations of coherence in sets of states. It is formulated around the construction of Bargmann scenarios, which specify relevant tuples of Bargmann invariants, and Bargmann polytopes, which bound the values that said invariants can have when the states are incoherent. We study their basic geometry, connect them to existing formalisms, and illustrate their physical relevance. Our construction opens new opportunities for the certification of quantum devices and lays the path toward a full quantum resource theory based entirely on multivariate traces of states.
Bargmann invariants, also known as multivariate traces of quantum states $\mathrm{Tr}({\ensuremath{\rho}}_{1}{\ensuremath{\rho}}_{2}\ensuremath{\cdots}{\ensuremath{\rho}}_{n})$, are unitary invariant quantities used to characterize weak values, Kirkwood-Dirac quasiprobabilities, out-of-time-order correlators, and geometric phases. Here we give a complete characterization of the set Bn{B}_{n}Bn​ of complex values that nthn\mathrm{th}nth-order invariants can take, resolving some recently proposed conjectures. We show that Bn{B}_{n}Bn​ is equal to the range of invariants arising from pure states described by Gram matrices of circulant form. We show that both ranges are equal to the nthn\mathrm{th}nth power of the complex unit nnn-gon and are therefore convex, which provides a simple geometric intuition. Finally, we show that any Bargmann invariant of order nnn is realizable using either qubit states or circulant qutrit states.
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.