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理论物理通讯：英文版

Knotted Picture of a Quantum Network of Two Nodes

QIANShang-Wu, GUZhi-Yu

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

This articlle discusses the variation of the knotted picture of the quantum pure state {X} = α[↑↓> + β[↑↓>with the wariation of the complex coeffieients α and β It is shown that there are three kinds of link that correspond to three different ranks of the matrix of covariance correlation tensor, i.e., the zero rank corresponds to trivial link, the rank one corresponds to the two-component link with two crossings, and the rank three corresponds to the two-component link with four crossings.

Related papers

Knotted Picture of a Quantum Network of Two Nodes

Qian Shang-Wu, Gu Zhi-Yu

This article discusses the variation of the knotted picture of the quantum pure state with the variation of the complex coefficients α and β. It is shown that there are three kinds of link that correspond to three different ranks of the matrix of covariance correlation tensor, i.e., the zero rank corresponds to trivial link, the rank one corresponds to the two-component link with two crossings, and the rank three corresponds to the two-component link with four crossings.

Communications in Theoretical Physics•Feb 15, 2004•6 citations

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

Gu Zhi-Yu, Qian Shang-Wu

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.

Chinese Physics B•Mar 31, 2015•0 citations

Knotted Picture of Degree of Entanglement of Quantum Pure State of Two Nodes （χ）=α（↑↓）＋β（↓↑）

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From the knotted pictures this article gives a possible quantitative measure of the degree of entanglement.We suggest to use the area ratio to measure the degree of entanglement, moreover, from the two parts of the non-overlapping area we can also know vividly and pictorially the phase difference between the two variable coefficients α and β.

理论物理通讯：英文版•Jan 1, 2004•0 citations

Knotted pictures of entangled states

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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.

Journal of Physics A Mathematical and General•Apr 12, 2002•9 citations

Knotted Picture of Degree of Entanglement of Quantum Pure State of Two Nodes | χ 〉 = α |↑↓〉 + β |↓↑〉

Gu Zhi-Yu, Qian Shang-Wu, Wang Jing-Shan

From the knotted pictures this article gives a possible quantitative measure of the degree of entanglement. We suggest to use the area ratio to measure the degree of entanglement, moreover, from the two parts of the non-overlapping area we can also know vividly and pictorially the phase difference between the two variable coefficients α and β .

Communications in Theoretical Physics•Apr 15, 2004•6 citations

Correspondence Between GHZ States and Oriented Links in Knot Theory

Qian Shang-Wu, Gu Zhi-Yu

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.

Communications in Theoretical Physics•Oct 15, 2002•9 citations
This articlle discusses the variation of the knotted picture of the quantum pure state {X} = α[↑↓> + β[↑↓>with the wariation of the complex coeffieients α and β It is shown that there are three kinds of link that correspond to three different ranks of the matrix of covariance correlation tensor, i.e., the zero rank corresponds to trivial link, the rank one corresponds to the two-component link with two crossings, and the rank three corresponds to the two-component link with four crossings.
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.
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.