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Communications in Theoretical Physics

Correspondence Between GHZ States and Oriented Links in Knot Theory

Qian Shang-Wu, Gu Zhi-Yu

Published Oct 15, 2002• 9 citations
Publisher

Abstract

This article generalizes our previous results for a quantum system of two nodes to a quantum system of nodes. It shows that there is a one to one correspondence between states for a quantum system of nodes and oriented links of the linkage N 1 in knot theory, where is the number of crossings of the link with m components, the subscript 1 denotes the order of the unoriented m components link with N crossings, i.e. the first type, actually the simplest type.

DOI
10.1088/0253-6102/38/4/421

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Communications in Theoretical Physics•Feb 15, 2004•6 citations
This article generalizes our previous results for a quantum system of two nodes to a quantum system of nodes. It shows that there is a one to one correspondence between states for a quantum system of nodes and oriented links of the linkage N 1 in knot theory, where is the number of crossings of the link with m components, the subscript 1 denotes the order of the unoriented m components link with N crossings, i.e. the first type, actually the simplest type.
This article generalizes our previous results for a quantum system of two nodes to a quantum system ofm(m ≥ 2) nodes. It shows that there is a one to one correspondence between 2m GHZ states for a quantum system ofm(m ≥ 2) nodes and 2m oriented links of the linkage N1 in knot theory, where N = 4 C2m = 2m(m - 1) is the numberof crossings of the link with m components, the subscript 1 denotes the order of the unoriented m components link withN crossings, i.e. the first type, actually the simplest type.
From the comparison of the correlation tensor in the theory of quantum networks and the Alexander relation matrix in the theory of knot crystals we find that there is a one-to-one correspondence between four Bell bases and four oriented links of the linkage 4 1 in knot theory. Meanwhile, we show that the inversion relations under the action of Pauli matrices are the same for the four Bell bases and their corresponding links. Similarly we show that there is also a one-to-one correspondence between 2 m GHZ states of m nodes and 2 m oriented links of m components.
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 two-node 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 4 1 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 submatrices 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.