Geometry of sets of Bargmann invariants

TL;DR

The paper investigates the boundaries of sets of Bargmann invariants, a class of unitary-invariant quantities in quantum theory. It develops a unified, dimension-independent framework to characterize the 3rd- and 4th-order invariants, proving the boundary description for the 4th order and confirming Fernandes et al.'s conjecture in that case, while proposing a generalization to arbitrary order $n$. Central to the approach are the numerical range for rank-one factors and the envelope method, which together show that $\mathcal{B}^ op_n(d)$ is a convex region with boundary described by the polar curve $r_n(\theta)=\cos^n\left(\frac{\pi}{n}\right)\sec^n\left(\frac{\theta-\pi}{n}\right)$, and that this boundary is attainable by a family of qubit pure states. The work points toward dimension-independence of these sets and opens avenues for applying Bargmann invariants to quantum imaginarity detection and related quantum-information tasks, with future work aimed at extending the results to general $n$ and exploring practical applications such as entanglement certification.

Abstract

Certain unitary-invariants, known as Bargmann invariants or multivariate traces of quantum states, have recently gained attention due to their applications in quantum information theory. However, determining the boundaries of sets of Bargmann invariants remains a theoretical challenge. In this study, we address the problem by developing a unified, dimension-independent formulation that characterizes the sets of the 3rd and 4th Bargmann invariants.In particular, our result for the set of 4th Bargmann invariants confirms the conjecture given by Fernandes \emph{et al.} [Phys.Rev.Lett.\href{https://doi.org/10.1103/PhysRevLett.133.190201}{\textbf{133}, 190201 (2024)}]. Based on the obtained results, we conjecture that the unified, dimension-independent formulation of the boundaries for sets of 3rd-order and 4th-order Bargmann invariants may extend to the general case of the $n$th-order Bargmann invariants. These results deepen our understanding of the fundamental physical limits within quantum mechanics and pave the way for novel applications of Bargmann invariants in quantum information processing and related fields.

Figures (1)

• Figure 1: (Color Online) The family of curves in polar form $r_n(\theta)=\cos^n(\frac{\pi}{n})\sec^n(\frac{\theta-\pi}{n})$, where $n\in\{3(\text{blue}),4(\text{brown}),5(\text{cyan}),6(\text{green}),7(\text{orange}),8(\text{purple}),9(\text{red})\}$. Note that $n\in\{3,4\}$, $r_n(\theta)=\cos^n(\frac{\pi}{n})\sec^n(\frac{\theta-\pi}{n})$ is indeed shown to be $\partial\mathcal{B}^\circ_n$ by Theorem \ref{['th:one']}. The black curve is the unit circle. Here the horizontal axis means the real part $\mathrm{Re}(\mathop{\mathrm{Tr}}\nolimits_{}\left(\psi_1\cdots\psi_n\right))$ and the vertical axis stands for the imaginary part $\mathrm{Im}(\mathop{\mathrm{Tr}}\nolimits_{}\left(\psi_1\cdots\psi_n\right))$.

Theorems & Definitions (22)

• Definition 1: Bargmann invariants for quantum states
• Definition 2: Numerical range of a complex square matrix
• Proposition 1
• Lemma 1
• proof
• Definition 3: Envelope, Bickel2020
• Theorem 1
• proof : Proof of Theorem \ref{['th:one']}
• Corollary 1
• proof