Many-body anticommutator and its applications in the normalization of completely symmetric states with Majorana's stellar representations

Understanding the quantum properties of many-body states is important in the fields of both quantum information and condensed-matter physics. For this purpose, we generalize the basic concept of the anticommutator from two operators to the many-body case. Some key properties are given for the many-body anticommutators, and examples are provided for Pauli operators and density matrices. Using these results and techniques from the symmetry group, in a straightforward way, we give the normalization of the completely symmetric states with Majorana's stellar representations. Our developed method will help to pave the way in the study of many-body symmetric systems.

Figure 1

The cycle diagram of $P=\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 2& 3& 1& 5& 4& 6\end{array}\right)$.

Figure 2

The Young diagrams associated with the partitions of $n=4$, where the rows stand for $l_i$, $i=1,...,q$, with $l_i\ge l_{i+1}$, as (a) 4, (b) $3+1$, (c) $2+2$, (d) $2+1+1$, and (e) $1+1+1+1$.