Geometry of sets of Bargmann invariants

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 3rd and 4th Bargmann invariants.In particular, our result for the set of 4th Bargmann invariants confirms the conjecture given by Fernandes \emph{et al.} [Phys.Rev.Lett.\href{https://doi.org/10.1103/PhysRevLett.133.190201}{\textbf{133}, 190201 (2024)}]. Based on the obtained results, we conjecture that the unified, dimension-independent formulation of the boundaries for sets of 3rd-order and 4th-order Bargmann invariants may extend to the general case of the $n$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.