Determination of W states equivalent under stochastic local operations and classical communication by their bipartite reduced density matrices with tree form
Xia Wu, Guojing Tian, Wei Huang, Qiaoyan Wen, Su‐Juan Qin, Fei Gao · Physical Review A · 2014
It has been known that, among pure states, $N$-qubit $W$ states cannot be uniquely determined by their arbitrary $(N\ensuremath{-}1)$ bipartite reduced density matrices. Parashar and Rana proved that among arbitrary states, $(N\ensuremath{-}1)$ 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 $(N\ensuremath{-}1)$ bipartite reduced density matrices, if the $(N\ensuremath{-}1)$ pairs of qubits constitute a tree graph on $N$ vertices, where each pair of qubits represents an edge.