Bargmann invariants, also known as multivariate traces of quantum states $\mathrm{Tr}({\rho }_1{\rho }_2\cdots {\rho }_n)$, 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.

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• Geometric & topological phases
• Quantum coherence & coherence measures
• Quantum correlations in quantum information
• Quantum correlations, foundations & formalism