Role of Covariance Correlation Tensor in Establishment of Criterion of Quantum Entanglement

Abstract

This article discusses the role of covariance correlation tensor in the establishment of the criterion of quantum entanglement. It gives a simple example to show the powerfulness in the treatment of quantum dense coding , and illustrates the fact that this method also provides theoretical basis for establishing corresponding knotted pictures.

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

“…For a quantum network of three nodes, the density operator ρ( j, k, l) is given by Refs. [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)…”

“…In our two previous papers, [17,18] we have discussed the correspondence between four Bell bases and the four oriented links of the linkage 4 1 in knot theory, [14][15][16] and the correspondence between GHZ states and the oriented links of the linkage 12 1 in knot theory. [19] Now we shall use the same method to find the correspondence of eight W states |W jkl and the eight expected oriented links in knot theory.…”

“…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 .…”

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

“…Recently, we have published quite a few articles about the knotted pictures of quantum states, quantum logic gates, and quantum processes, [4−10] and have successfully demonstrated the intimate relationship between knot theory and quantum entanglement. Based on the studies about the correspondence between the Bell bases and the linkage 4 1 in the knot theory, [4,5] we have shown that there exists a very simple and vivid way of obtaining the linkage 4 1 by using the torus knot theory in Ref. [11], where we firstly found the knotted picture of Bell bases (n = 2, n is the number of qubits) on the surface of a trivial torus and the torus link K 4,2 .…”

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

“…[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. Furthermore, we have given the knotted picture of the complete quantum measurement process of quantum teleportation.…”