Numerical ranges of Bargmann invariants
Jianwei Xu
TL;DR
This work resolves the long-standing question of the physically realizable values of Bargmann invariants in finite-dimensional quantum systems by proving that all admissible values are achievable either with pure states whose Gram matrices are circular or with qubit states. The authors connect the problem to Gram-matrix representations, establish key convexity and closure properties, and show that for $n\ge 3$ and $d\ge 2$ the numerical ranges coincide with the qubit/circular Gram region $\mathcal{B}_{n,c}$, i.e., $\boldsymbol{B}_{n,d}=\mathcal{B}_{n,d}=\boldsymbol{B}_{n,2}=\mathcal{B}_{n,2}=\mathcal{B}_{n,c}$ (with $\mathcal{B}_{2}=[0,1]$). This yields explicit boundary descriptions and simplifies high-dimensional behavior to representative low-dimensional cases, delivering a rigorous mathematical foundation for the use of Bargmann invariants in quantum information tasks like geometric phase analysis, state discrimination, and error mitigation. The results provide a precise, operational understanding of the achievable values and facilitate applications across quantum information processing.
Abstract
Bargmann invariants have recently emerged as powerful tools in quantum information theory, with applications ranging from geometric phase characterization to quantum state distinguishability. Despite their widespread use, a complete characterization of their physically realizable values has remained an outstanding challenge. In this work, we provide a rigorous determination of the numerical range of Bargmann invariants for quantum systems of arbitrary finite dimension. We demonstrate that any permissible value of these invariants can be achieved using either (i) pure states exhibiting circular Gram matrix symmetry or (ii) qubit states alone. These results establish fundamental limits on Bargmann invariants in quantum mechanics and provide a solid mathematical foundation for their diverse applications in quantum information processing.