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Communications in Theoretical Physics

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

Published Apr 15, 2004• 6 citations
Publisher

Abstract

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 .

DOI
10.1088/0253-6102/41/4/531

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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 .
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 degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings [A. Shimony, Ann. NY. Acad. Sci. 755, p.675 (1995) and H. Barnum and N. Linden, J. Phys. A: Math. Gen. 34, p.6787 (2001)], is explored for bipartite and multipartite pure and mixed states. It is determined for arbitrary two-qubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states.