On the correspondence between three nodes W states in quantum network theory and the oriented links in knot theory

• Published in: Chinese physics B 2015-04, Vol.24 (4), p.36-41
• Main Author: 顾之雨 钱尚武
• Format: Article
• Language: English
• Subjects: Alexander relation matrix; covariance correlation tensor; Crystal structure; Knot theory; Knots; knotted picture; Links; Mathematical analysis; Networks; Pictures; quantum entangled state; Qubits (quantum computing); W states; 协方差相关张量; 取向; 对应关系; 纽结理论; 网络理论; 节点; 量子纠缠态
• ISSN: 1674-1056; 2058-3834; 1741-4199
• DOI: 10.1088/1674-1056/24/4/040301

The GHZ states and W states are two fundamental types of three qubits quantum entangled states. For finding the knotted pictures of three nodes W states, on the one side, we empty any one node, thus obtaining three degenerated twonode W states, then we find the nonzero submatrix of the corresponding covariance correlation tensor in quantum network theory. On the other side, excepting the linkage 41 corresponding to Bell bases, we conjecture that the another one possible oriented link （which is composed of two-component knots entangled with each other and has four crossings） would be the required knotted pictures of the two nodes W states, thence obtain the nonzero submatrix of the Alexander relation matrix in the theory of knot crystals for these knotted pictures. The equality of the two nonzero submatfices of different kinds thus verify the exactness of our conjecture. The superposition of three knotted pictures of two-node W states from different choices of the emptied node gives the knotted pictures of three-node W states, thus shows the correspondence between three-node W states in quantum network theory and the oriented links in knot theory. Finally we point out that there is an intimate and simple relationship between the knotted pictures of GHZ states and W states.