Elementary characterization of Bargmann invariants

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 ${B}_{n}$ of complex values that $n\mathrm{th}$-order invariants can take, resolving some recently proposed conjectures. We show that ${B}_{n}$ 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 $n\mathrm{th}$ power of the complex unit $n$-gon and are therefore convex, which provides a simple geometric intuition. Finally, we show that any Bargmann invariant of order $n$ is realizable using either qubit states or circulant qutrit states.