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Quantum entanglement, as an indispensable resource in quantum communication and quantum computation, is widely used in the field of quantum information. However, people's understanding on entanglement is quite limited both theoretically and experimentally. How to determine whether a given quantum state is entangled is still an important task. The entanglement witness is a kind of special self-adjoint operator, it can be used to determine whether a quantum state is an entangled state. This provides a new direction for the determination of entangled states. Entanglement witness has its own unique characteristics in various kinds of entanglement criterion. It is the most effective tool for detecting multipartite entanglement, and the most useful method to detect entanglement in experiments. In the background of quantum theory, we use theory of operators to make a thorough and systematic study of the construction of entanglement witness in this paper. First, from the definition of an entanglement witness, a general method is given to construct an entanglement witness. It is proved that when the maximal expectation CA of an observable A in the separable pure states is strictly less than its biggest eigenvalue max(A), the operator WC=CI-A is an entanglement witness provided that CA C max(A). Although the entanglement witness WCA can detect more entangled states than WC, but it is difficult to calculate the exact value of CA, and the estimate of the upper bound of CA is easier. Therefore, it is more convenient to construct entanglement witness WC than WCA. In quantum computation, a graph state is a special kind of multi-qubit state that can be represented by a graph. Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits. Graph states play a crucial role in many applications of quantum information theory, such as quantum error correcting codes, measurement-based quantum computation, and quantum simulation. Consequently, a significant effort is devoted to the creation and investigation of graph states. In the last part of this paper, as applications of our method, a series of methods for constructing an entanglement witness is obtained in the stabilizer formalism. It is also proved that how entanglement witnesses can be derived for a given graph state, provided some stabilizing operators of the graph state are known. Especially, when A is made up of some stabilizing operators of a graph state, entanglement witness WCA becomes one in literature.
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Keywords:
- entanglement witness /
- graph state /
- stabilizer
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[25] Jafarizadeh M A, Najarbashi G, Habibian H 2007 Phys. Rev. A 75 052326
[26] Jafarizadeh M A, Sufiani R, Nami S, Golmohammadi M 2012 Quantum. Inf. Process. 11 729
[27] Cheng S, Chen J, Wang L 2017 Physics 46 416 (in Chinese) [程嵩, 陈靖, 王磊 2017 物理 46 416]
[28] Deng D L, Li X P, Sarma S D 2017 Phys. Rev. X 7 021021
[29] Levine Y, Yakira D, Cohen N, Shashua A 2017 arXiv: 1704.01552
[30] Carleo G, Troyer M 2017 Science 355 602
[31] Gao X, Duan L M 2017 Nature Commun. 8 662
[32] Tth G, Ghne O, Briegel H J 2005 Phys. Rev. Lett. 95 120405
[33] Hein M, Eisert J, Briegel H J 2003 Phys. Rev. A 69 062311
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[1] Bennett C H, Brassard G, Crpeau C, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895
[2] Ekert A K 1991 Phys. Rev. Lett. 67 661
[3] Steane A 1998 Rep. Prog. Phys. 61 117
[4] Mattle K, Weinfurter H, Kwiat P G, Zeilinger A 1996 Phys. Rev. Lett. 76 4656
[5] Hillery M, Buvek V, Berthiaume A 1999 Phys. Rev. A 59 1829
[6] Long G L, Liu X S 2002 Phys. Rev. A 65 032302
[7] Sheng Y B, Zhou L 2017 Sci. Bull. 62 1025
[8] Deng F G, Ren B C, Li X H 2017 Sci. Bull. 62 46
[9] Cong M Y, Yang J, Huang Y X 2016 Acta Phys. Sin. 65 170301 (in Chinese) [丛美艳, 杨晶, 黄燕霞 2016 65 170301]
[10] Ren B C, Deng F G 2015 Acta Phys. Sin. 64 160303 (in Chinese) [任宝藏, 邓富国 2015 64 160303]
[11] Zong X L, Yang M 2016 Acta Phys. Sin. 65 080303 (in Chinese) [宗晓岚, 杨名 2016 65 080303]
[12] Yang F, Cong S 2011 Chin. J. Quant. Elect. 28 391 (in Chinese) [杨霏, 丛爽 2011 量子电子学报 28 391]
[13] Lewenstein M, Kraus B, Cirac J I, Horodecki P 2000 Phys. Rev. A 62 052310
[14] Lewenstein M, Kraus B, Horodecki P, Cirac J I 2001 Phys. Rev. A 63 044304
[15] Tth G, Ghne O 2005 Phys. Rev. Lett. 94 060501
[16] Ghne O, Hyllus P, Bruss D, Ekert A, Lewenstein M, Macchiavello C, Sanpera A 2002 Phys. Rev. A 66 062305
[17] Tth G 2004 Phys. Rev. A 69 052327
[18] Brukner C, Vedral V, Zeilinger A 2006 Phys. Rev. A 73 012110
[19] Wu L A, Bandyopadhyay S, Sarandy M S, Lidar D A 2005 Phys. Rev. A 72 032309
[20] Tth G, Ghne O 2005 Phys. Rev. A 72 022340
[21] Doherty A C, Parrilo P A, Spedalieri F M 2005 Phys. Rev. A 71 032333
[22] Vianna R O, Doherty A C 2006 Phys. Rev. A 74 052306
[23] Jafarizadeh M A, Rezaee M, Yagoobi S K A S 2005 Phys. Rev. A 72 062106
[24] Jafarizadeh M A, Rezaee M, Ahadpour S 2006 Phys. Rev. A 74 042335
[25] Jafarizadeh M A, Najarbashi G, Habibian H 2007 Phys. Rev. A 75 052326
[26] Jafarizadeh M A, Sufiani R, Nami S, Golmohammadi M 2012 Quantum. Inf. Process. 11 729
[27] Cheng S, Chen J, Wang L 2017 Physics 46 416 (in Chinese) [程嵩, 陈靖, 王磊 2017 物理 46 416]
[28] Deng D L, Li X P, Sarma S D 2017 Phys. Rev. X 7 021021
[29] Levine Y, Yakira D, Cohen N, Shashua A 2017 arXiv: 1704.01552
[30] Carleo G, Troyer M 2017 Science 355 602
[31] Gao X, Duan L M 2017 Nature Commun. 8 662
[32] Tth G, Ghne O, Briegel H J 2005 Phys. Rev. Lett. 95 120405
[33] Hein M, Eisert J, Briegel H J 2003 Phys. Rev. A 69 062311
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