Knotted pictures of entangled states

Knotted pictures of the Bell bases on the surface of a trivial torus

“…[4−6] By comparing the covariance correlation tensor in the theory of quantum networks and the Alexander relation matrix in the theory of knot crystals, we have found that there is a one-to-one correspondence between four Bell bases and four oriented links of the linkage 4 1 in knot theory. [4,5] We have also found a one-to-one correspondence between GHZ states and the oriented links. [6] Hence, we have used the classical language of knot theory to describe the property of the algebraic structure of quantum entanglement, and revealed the interrelation of two seemingly different phenomena.…”

“…Its significance and application are still in the searching and progressing state. Based on our previous studies on the correspondence between Bell bases and the linkage 4 1 in knot theory, [4,5] in this paper we shall show that there exists a very simple and clear way of obtaining the linkage 4 1 by using the method of torus knot theory. The significance of this work is that by means of this method we can easily study the complex problems about the correspondences between n-qubit (n > 2) quantum states and their corresponding knotted pictures.…”

“…It is well known that for the system consisting of two spin -1/2 particles 1 and 2, there are four wellknown entangled Bell bases, which are [4,5]…”

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

“…[13], [20], [21], and [22], for brevity we will not list this expression, here we only point out that equation ( 4) can be directly obtained from ρ( j, k, l) by emptying the first node j, hereafter we shall use the symbols j, k, l to represent the quantities related to the first, the second, and the third nodes respectively, hence the density operators corresponding to three degenerated two-nodes quantum states ρ(k, l), ρ( j, l), ρ( j, k) can be directly obtained from ρ( j, k, l) by emptying j, k, l respectively. Equation ( 4) can be written as [13,17,20,21] ρ(k, l) = ρ(k) ⊗ ρ(l)…”

“…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 .…”