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Chin. Phys. B, 2015, Vol. 24(4): 040301    DOI: 10.1088/1674-1056/24/4/040301
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On the correspondence between three nodes W states in quantum network theory and the oriented links in knot theory

Gu Zhi-Yu (顾之雨)a, Qian Shang-Wu (钱尚武)b
a Physics Department, Capital Normal University, Beijing 100048, China;
b School of Physics, Peking University, Beijing 100871, China
Abstract  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 41 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.
Keywords:  quantum entangled state      knotted picture      W states      covariance correlation tensor      Alexander relation matrix  
Received:  17 September 2014      Revised:  02 November 2014      Accepted manuscript online: 
PACS:  03.65.Ud (Entanglement and quantum nonlocality)  
  03.67.Hk (Quantum communication)  
  02.10.Kn (Knot theory)  
Corresponding Authors:  Qian Shang-Wu     E-mail:  swqian@pku.edu.cn

Cite this article: 

Gu Zhi-Yu (顾之雨), Qian Shang-Wu (钱尚武) On the correspondence between three nodes W states in quantum network theory and the oriented links in knot theory 2015 Chin. Phys. B 24 040301

[1] Ekert A and Josza R 1996 Rev. Mod. Phys. 68 733
[2] DiVincenzo D P 1995 Science 270 255
[3] Huches C H, Gisin N, Griffiths R B, Niu C S and Peres A 1997 Phys. Rev. A 56 1163
[4] Horodecki M and Horodecki P 1999 Phys. Rev. A 59 4206
[5] Cerf N J, Adami C and Gingrich R M 1999 Phys. Rev. A 60 898
[6] Hardy L and Song D D 2000 Phys. Rev. A 62 052315
[7] Fan H 2001 Phys. Lett. A 286 81
[8] Thapliyal A V 1999 Phys. Rev. A 59 3336
[9] Vedral V, Plenio M B, Rippin M A and Knight P L 1997 Phys. Rev. Lett. 78 2275
[10] Horodecki P and Lewenstein M 2000 Phys. Rev. Lett. 85 2657
[11] Bennett C H, DiVincenzo D P, Smolin J A and Wootters W K 1996 Phys. Rev. A 54 3824
[12] Dür W, Vidal G and Cirac J I 2000 Phys. Rev. A 62 062314
[13] Mahler G and Weberrufz V A 1998 Quantum Networks, 2nd edn. (revised and enlarged) (Berlin: Springer)
[14] Kauffman L H 1993 Knots And Physics, 2nd ed,. (Singapore: World Scientific)
[15] Kleinert H 1990 Path Integrals in Quantum Mechanics and Polymer Physics (Singapore: World Scientific)
[16] Yang C N and Ge M L (eds.) 1989 Braid Group, Knot Theory and Statistical Mechanics (New Jersey: World Scientific)
[17] Qian S W and Gu Z Y 2002 J. Phys. A: Math. Gen. 35 3733
[18] Qian S W and Gu Z Y 2002 Commun. Theor. Phys. 37 659
[19] Qian S W and Gu Z Y 2002 Commun. Theor. Phys. 38 421
[20] Qian S W and Gu Z Y 2003 Commun. Theor. Phys. 39 15
[21] Gu Z Y and Qian S W 2003 Commun. Theor. Phys. 39 421
[22] Gu Z Y and Qian S W 2003 Commun. Theor. Phys. 40 33
[23] Qian S W and Gu Z Y 2004 Commun. Theor. Phys. 41 201
[24] Gu Z Y and Qian S W 2004 Commun. Theor. Phys. 41 531
[25] Gu Z Y and Qian S W 2008 Commun. Theor. Phys. 49 65
[26] Gu Z Y and Qian S W 2008 Commun. Theor. Phys. 49 1163
[27] Gu Z Y and Qian S W 2009 Commun. Theor. Phys. 51 769
[28] Gu Z Y and Qian S W 2009 Commun. Theor. Phys. 51 967
[29] Gu Z Y and Qian S W 2010 Chin. Phys. B 19 080306
[30] Gu Z Y and Qian S W 2011 Chin. Phys. B 20 090201
[31] Gu Z Y and Qian S W 2012 Chin. Phys. B 21 070201
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