An Elementary Characterization of Bargmann Invariants

TL;DR

The paper characterizes the full range of complex values $B_n$ of $n$-th order Bargmann invariants, showing it equals the range from pure states with circulant Gram matrices and equals the $n$-th power of the regular $n$-gon, $\mathcal{P}_n^n$. It proves $\mathcal{R}_{n\mid\mathrm{circ}}=\mathcal{P}_n$ and that $\mathcal{R}_n=\mathcal{R}_{n\mid\mathrm{circ}}$, whence $B_n=\mathcal{B}_{n\mid\mathrm{circ}}=\mathcal{P}_n^n$. Any Bargmann invariant can be realized using only qubits or circulant qutrits, providing a practical experimental route. These results yield geometric bounds on negativity/imaginarity in Kirkwood-Dirac quasiprobabilities, constrain higher-order OTOCs, and inform geometric-phase experiments and multiphoton indistinguishability analyses.

Abstract

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.

Figures (2)

• Figure 1: Characterization of $n$-th order Bargmann invariants, illustrated here for $n=5$, with the complex unit circle as reference. a) The range of values that pure Bargmann invariants of order $n$ can take, $B_{n}$, is a teardrop-shaped convex region in the complex plane (line-shaded region). b) The values that the $n$-th complex roots of Bargmann invariants of order $n$ can take, $\mathcal{R}_{n}$, are the filled unit $n$-gon, $\mathcal{P}_n$. We prove this first for circulant tuples of states. The vertices, edges and interior of $\mathcal{P}_n$ are roots of Bargmann invariants realizable by states with dimensions 1, 2 (qubits) and 3 (qutrits), respectively, with circulant Gram matrices. The principal roots of Bargmann invariants (line-shaded triangle) are always interior to the three vertices $1$, $\omega_n$ and $y_n = \omega_n^{\lceil n/2\rceil}$, which imples that any Bargmann invariant may be realized by a circulant tuple of qutrit states.
• Figure 2: Illustration of Theorem \ref{['thm:main_theorem']} for $n=5$. Given a tuple of states $V=(\ket{v_1},\ldots,\ket{v_n})$ we can use gauge freedom to map the inner products $\braket{v_1}{v_2}, \ldots, \braket{v_n}{v_1}$ (black triangles) onto the same complex ray, without changing the corresponding Bargmann invariant $\Delta_V$. The geometric mean of the inner products (open circle), which corresponds to the $n$-the root of their Bargmann invariant, must lie closer to the origin than their arithmetic mean, $\Delta^{1/n}_\text{circ}$ (filled circle). Both the origin and $\Delta^{1/n}_\text{circ}$ are roots of Bargmann invariants that are realizable by circulant tuples of states. Since the set $\mathcal{R}_{n\mid\mathrm{circ}}$ is convex, $\Delta_V$ must be realizable as a Bargmann invariant of a circulant tuple of states.

Theorems & Definitions (14)

• Theorem 1
• proof
• Corollary 2
• proof
• Corollary 3
• proof
• Corollary 4
• proof
• Theorem 5
• proof