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Elementary characterization of Bargmann invariants
Authors: Sagar Silva Pratapsi; João Gouveia; Leonardo Novo; Ernesto F. Falcão
Ref.: Phys. Rev. A 112, 042421 (2025)
Abstract: Bargmann invariants, also known as multivariate traces of quantum states Tr(ρ1ρ2· · · ρn ), 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 Bn of complex values that nth-order invariants can take, resolving some recently proposed conjectures. We show that Bn 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 nth 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.
