An Elementary Characterization of Bargmann Invariants
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911210530668544 |
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| author | Pratapsi, Sagar Silva Gouveia, João Novo, Leonardo Galvão, Ernesto F. |
| author_facet | Pratapsi, Sagar Silva Gouveia, João Novo, Leonardo Galvão, Ernesto F. |
| contents | Bargmann invariants, also known as multivariate traces of quantum states $\operatorname{Tr}(ρ_1 ρ_2 \cdots ρ_n)$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17132 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Elementary Characterization of Bargmann Invariants Pratapsi, Sagar Silva Gouveia, João Novo, Leonardo Galvão, Ernesto F. Quantum Physics Bargmann invariants, also known as multivariate traces of quantum states $\operatorname{Tr}(ρ_1 ρ_2 \cdots ρ_n)$, 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. |
| title | An Elementary Characterization of Bargmann Invariants |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2506.17132 |