On the Bargmann invariants for quantum imaginarity

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. Galv\~ao, 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 $n$ quantum states. Then, we study the properties of the relevant Bargmann invariant set $\mathcal{B}_n$ 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.