Determination of $W$ states equivalent under stochastic local operations and classical communication by their bipartite reduced density matrices with tree form

It has been known that, among pure states, $N$-qubit $W$ states cannot be uniquely determined by their arbitrary $(N-1)$ bipartite reduced density matrices. Parashar and Rana proved that among arbitrary states, $(N-1)$ bipartite reduced density matrices that the pairs of qubits constitute a star graph or a line graph can uniquely determine stochastic local operations and classical communication (SLOCC) equivalent $W$ states, and we generalize this conclusion into tree graph. In this paper, we show that all SLOCC equivalent $W$ states can be uniquely determined (among pure, mixed states) by their $(N-1)$ bipartite reduced density matrices, if the $(N-1)$ pairs of qubits constitute a tree graph on $N$ vertices, where each pair of qubits represents an edge.

Figure 1

Coefficient of $|i_s{i}_t\rangle \langle j_s{j}_t|$ in ${\rho }_{\psi }^{st}$. Here we show part of the tree graph on $N$ vertices, $\{s_1,...,s_n,s,t,t_1,...,t_n\}\subseteq \{1,2,...,N\}$, the vertices that do not appear in this figure also vary over 0 or 1.

Figure 2

Coefficient of $|01\rangle \langle 01|$ in ${\rho }_{\psi }^{st}$.

Figure 3

Coefficient of $|01\rangle \langle 10|$ in ${\rho }_{\psi }^{st}$.

Figure 4

Coefficient of $|01\rangle \langle 01|$ in ${\rho }_{\psi }^{st}$.

Figure 5

Coefficient of $|10\rangle \langle 10|$ in ${\rho }_{\psi }^{st}$.