Knotted pictures of the GHZ states on the surface of a trivial torus

Abstract

By means of the torus knot theory method, this paper presents the complete process of obtaining the knotted pictures of eight GHZ states on the surface of a trivial torus from the knotted pictures of eight basic three-qubit states on the surface of a trivial torus. Thus, we obtain eight knotted pictures 121 linkage on the ordinary plane.

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

“…We have already discovered the knotted pictures of GHZ states ({|GHZ jkl }). [19,31] When we empty any one node, then the obtained results {|GHZ kl }, {|GHZ jl }, {|GHZ jk } are just the correponding knotted pictures of the Bell bases, which have the largest degree of entanglement of the linkage 4 1 . The linkage 4 1 is a collection of two component knots, [17] it has four crossings when one component knot j (its ring parameter is s, for oriented knot, s + and s − denote the anticlockwise and clockwise directions of this knot respectively [17] ) crosses another component knot k (its ring parameter is t, for an oriented knot, t + and t − denote the anticlockwise and clockwise directions of this knot 040301-3 respectively [17] ).…”

“…[17] Furthermore, when we apply the flip on the right three negative crossings, then figure 4 becomes the knotted picture of one of the eight GHZ states. [19,31] Hence we can from the knotted pictures of GHZ states obtain those of W states just by the application of the surgical operation flip, or vice versa. This is the very important and very simple interrelations of the two fundamental types of the 3-node entangled quantum states: GHZ state and W state.…”

“…Because of the very importance of quantum entanglement [1][2][3][4][5][6][7][8][9][10][11][12][13] we try to find the connection between entangled states and oriented links in knot theory, [14][15][16] and have successfully found such deep connections. [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] Recently, based on our previous works about the correspondence between four Bell bases and four oriented links 4 1 in knot theory, [17,18] and the correspondence of eight GHZ quantum entangled states and eight oriented links 12 1 in knot theory, [19] furthermore, we have shown that there exists a very simple and vivid way of obtaining the linkage 4 1 by using the method of torus knot theory. [30] Firstly we find the knotted picture of Bell bases (number of qubits m = 2) on the surface of trivial torus, the torus link K 4,2 , then after quitting the trivial torus and projecting the torus link on a plane, we can easily obtain the knotted pictures of Bell bases, the linkage 4 1 .…”

“…Similarly on the ordinary plane, [30] we find the knotted picture of eight GHZ quantum states (number of qubits m = 3) on the surface of trivial torus, the torus link K 6,3 , then after quitting the trivial torus and projecting the torus link on a plane, we can easily obtain the knotted pictures of eight entangled GHZ quantum states, and the linkage 12 1 , on the ordinary plane. [31] In general, for m qubits GHZ states we have shown that the knotted picture of 2 m oriented links can be represented by the torus link K 2m,m on the surface of trivial torus. [31] From all our research about the relationship between entangled states and oriented links we can confidently draw the following conclusions: there ex-ists the correspondence between entangled quantum states and oriented links in knot theory; every qubit corresponds to its own knot in the oriented link corresponding to the given entangled state; this correspondence can be used to predict some new undiscovered entangled states and related phenomena.…”

“…[31] In general, for m qubits GHZ states we have shown that the knotted picture of 2 m oriented links can be represented by the torus link K 2m,m on the surface of trivial torus. [31] From all our research about the relationship between entangled states and oriented links we can confidently draw the following conclusions: there ex-ists the correspondence between entangled quantum states and oriented links in knot theory; every qubit corresponds to its own knot in the oriented link corresponding to the given entangled state; this correspondence can be used to predict some new undiscovered entangled states and related phenomena.…”