Majorana's stellar representation of single-particle reduced density matrix for completely symmetric states

The completely symmetric states play an essential role in quantum physics. In this paper, we calculate the reduced density matrix (RDM) for a single particle of the completely symmetric system coupled by $N$ spin-$\frac{1}{2}$ particles, because it helps to investigate the evolution of expectation value for the observable and to calculate the entanglement between the subsystems. Furthermore, we use Majorana's stellar representation (MSR) to represent the results because it provides an intuitive geometric perspective to comprehend the quantum states in the high-dimensional Hilbert space with distributions and trajectories of the Majorana stars on a Bloch sphere. With the operation properties of the generalized many-body anticommutator, we get the general MSR form of a single-qubit RDM. As the application and verification, we calculate the single-qubit RDM for the Dicke states with the results. Similarly, we further solve the RDM of the spin-$\frac{N}{2}$ state in a uniform magnetic field and study the systems with symmetric structures on the Bloch sphere. The results exhibit the relations between the composite systems and the subsystems, and provide a new idea for the numerical solution of multiqubit systems.