Numerical ranges of Bargmann invariants

TL;DR

This paper rigorously characterizes the numerical range of Bargmann invariants in finite-dimensional quantum systems, revealing that all permissible values can be realized with specific pure or qubit states, thus establishing fundamental quantum limits.

Contribution

It provides a complete mathematical characterization of the numerical range of Bargmann invariants for arbitrary finite-dimensional quantum systems, a previously unresolved challenge.

Findings

Any permissible Bargmann invariant value can be achieved with pure states with circular Gram matrix symmetry.

Qubit states alone can realize all permissible invariant values.

The results establish fundamental limits on Bargmann invariants in quantum mechanics.

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.