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arXiv · 2506.17132

An Elementary Characterization of Bargmann Invariants

Sagar Silva Pratapsi·João Gouveia·Leonardo Novo·Ernesto F. Galvão
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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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Sagar Silva Pratapsi, João Gouveia, Leonardo Novo, Ernesto F. Galvão. 2025-06-20. An Elementary Characterization of Bargmann Invariants. https://doi.org/10.1103/hsnv-wpt3

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