An Elementary Characterization of Bargmann Invariants

• URL: http://arxiv.org/abs/2506.17132v2
• Date: Fri, 04 Jul 2025 17:05:20 GMT
• Title: An Elementary Characterization of Bargmann Invariants
• Authors: Sagar Silva Pratapsi, João Gouveia, Leonardo Novo, Ernesto F. Galvão,
• Abstract summary: We give a complete characterization of the set $B_n$ of complex values that $n$-th order invariants can take.<n>We show that both ranges are equal to the $n$-th power of the complex unit $n$-gon, and are therefore convex.
• Score: 0.6749750044497732
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Bargmann invariants, also known as multivariate traces of quantum states $\operatorname{Tr}(\rho_1 \rho_2 \cdots \rho_n)$, are unitary invariant quantities used to characterize weak values, Kirkwood-Dirac quasiprobabilities, out-of-time-order correlators (OTOCs), and geometric phases. Here we give a complete characterization of the set $B_n$ of complex values that $n$-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 $n$-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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