An Elementary Characterization of Bargmann Invariants

TL;DR

This paper provides a complete geometric characterization of Bargmann invariants, revealing their range as the n-th power of the complex unit n-gon and demonstrating realizability with qubit or circulant qutrit states.

Contribution

It offers a full characterization of the set of possible Bargmann invariant values, resolving conjectures and linking them to geometric shapes and specific quantum state classes.

Findings

Range of invariants equals the n-th power of the complex unit n-gon.

Range is convex and geometrically intuitive.

Any invariant can be realized with qubit or circulant qutrit states.

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.