Top

Quantum Information Processing

Published in:

01-02-2017

Quantum teleportation and Birman–Murakami–Wenzl algebra

Authors: Kun Zhang, Yong Zhang

Published in: Quantum Information Processing | Issue 2/2017

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this paper, we investigate the relationship of quantum teleportation in quantum information science and the Birman–Murakami–Wenzl (BMW) algebra in low-dimensional topology. For simplicity, we focus on the two spin-1/2 representation of the BMW algebra, which is generated by both the Temperley–Lieb projector and the Yang–Baxter gate. We describe quantum teleportation using the Temperley–Lieb projector and the Yang–Baxter gate, respectively, and study teleportation-based quantum computation using the Yang–Baxter gate. On the other hand, we exploit the extended Temperley–Lieb diagrammatical approach to clearly show that the tangle relations of the BMW algebra have a natural interpretation of quantum teleportation. Inspired by this interpretation, we construct a general representation of the tangle relations of the BMW algebra and obtain interesting representations of the BMW algebra. Therefore, our research sheds a light on a link between quantum information science and low-dimensional topology.

previous article Quantum broadcast scheme and multi-output quantum teleportation via four-qubit cluster state

next article Violation of Bell inequalities from symmetry: the three orbits case

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

The Bell transform \(B_{\textit{ell}}\) in this paper is defined as

where \(k=k(k^\prime ,l^\prime )\) and \(l=l(k^\prime ,l^\prime )\) are bijective functions of \(k^\prime \) and \(l^\prime \), respectively; \(e^{i \phi _{\textit{kl}}}\) is the phase factor; and \(S_{kl}\) and \(Q_{kl}\) are single-qubit gates. Such a definition of the Bell transform differs from the proposed definition of the Bell transform in the previous research [34] where single-qubit gates \(S_{kl}\) and \(Q_{kl}\) are not involved.

Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, UK (2000 and 2011)

Preskill, J.: Lecture Notes on Quantum Computation. http://www.theory.caltech.edu/preskill

Bennett, C.H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen Channels. Phys. Rev. Lett. 70, 1895 (1993)ADSMathSciNetCrossRefMATH

Vaidman, L.: Teleportation of quantum states. Phys. Rev. A 49, 1473–1475 (1994)ADSMathSciNetCrossRef

Braunstein, S.L., D’Ariano, G.M., Milburn, G.J., Sacchi, M.F.: Universal teleportation with a twist. Phys. Rev. Lett. 84, 3486–3489 (2000)ADSCrossRef

Werner, R.F.: All teleportation and dense coding schemes. J. Phys. A Math. Theor. 35, 7081–7094 (2001)ADSMathSciNetMATH

Aravind, P.K.: Borromean Entanglement of the GHZ State. Potentiality, Entanglement and Passion-at-a-Distance. Springer, Berlin (1997)CrossRefMATH

Kauffman, L.H., Lomonaco Jr., S.J.: Quantum entanglement and topological entanglement. New J. Phys. 4, 73 (2002)ADSMathSciNetCrossRef

Kauffman, L.H.: Knots and Physics. World Scientific Publishers, Singapore (2002)MATH

10.

Yang, C.N.: Some exact results for the many body problems in one dimension with repulsive delta function interaction. Phys. Rev. Lett. 19, 1312–1314 (1967)ADSMathSciNetCrossRefMATH

11.

Baxter, R.J.: Partition function of the eight-vertex lattice model. Ann. Phys. 70, 193–228 (1972)ADSMathSciNetCrossRefMATH

12.

Perk, J.H.H., Au-Yang, H.: Yang–Baxter Equations, Encyclopedia of Mathematical Physics. Elsevier, Oxford (2006)

13.

Dye, H.: Unitary solutions to the Yang–Baxter equation in dimension four. Quantum Inf. Process. 2, 117–150 (2003)MathSciNetCrossRefMATH

14.

Kauffman, L.H., Lomonaco Jr., S.J.: Braiding operators are universal quantum gates. New J. Phys. 6, 134 (2004)ADSCrossRef

15.

Zhang, Y., Kauffman, L.H., Ge, M.L.: Universal Quantum Gate, Yang–Baxterization and Hamiltonian. Int. J. Quantum Inf. 4, 669–678 (2005)CrossRefMATH

16.

Alagic, G., Jarret, M., Jordan, S.P.: Yang-Baxter operators need quantum entanglement to distinguish knots. J. Phys. A Math. Theor. 49, 075203 (2016)ADSMathSciNetCrossRefMATH

17.

Temperley, H.N.V., Lieb, E.H.: Relations between the percolation and colouring problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the percolation problem. Proc. R. Soc. A 322, 251 (1971)ADSMathSciNetCrossRefMATH

18.

Birman, J.S., Wenzl, H.B.: Link polynomials and a new algebra. Trans. Am. Math. Soc. 313, 249–273 (1989)MathSciNetCrossRefMATH

19.

Murakami, J.: The Kauffman polynomial of links and representation theory. Osaka J. Math. 24, 745–758 (1987)MathSciNetMATH

20.

Jones, V.: On a certain value of the Kauffman polynomial. Commun. Math. Phys. 125, 459–467 (1989)ADSMathSciNetCrossRefMATH

21.

Cheng, Y., Ge, M.L., Xue, K.: Yang–Baxterization of braid group representations. Commun. Math. Phys. 136, 195–208 (1991)ADSMathSciNetCrossRefMATH

22.

Wang, G., Xue, K., Sun, C., Zhou, C., Hu, T., Wang, Q.: Temperley–Lieb algebra, Yang–Baxterization and universal gate. Quantum Inf. Process. 9, 699–710 (2010)MathSciNetCrossRefMATH

23.

Zhang, Y.: Teleportation, braid group and Temperley–Lieb algebra. J. Phys. A Math. Theor. 39, 11599–11622 (2006)ADSMathSciNetMATH

24.

Zhang, Y., Kauffman, L.H.: Topological-Like features in diagrammatical quantum circuits. Quantum Inf. Process. 6, 477–507 (2007)CrossRefMATH

25.

Zhang, Y.: Braid group, Temperley–Lieb algebra, and quantum information and computation. AMS Contemp. Math. 482, 52 (2009)MathSciNetMATH

26.

Zhang, Y., Zhang, K., Pang, J.-L.: Teleportation-based quantum computation, extended Temperley–Lieb diagrammatical approach and Yang–Baxter equation. Quantum Inf. Process. 15, 405–464 (2016)ADSMathSciNetCrossRefMATH

27.

Wang, G., Xue, K., Sun, C., Liu, B., Liu, Y., Zhang, Y.: Topological basis associated with B-M-W algebra: two spin-1/2 realization. Phys. Lett. A 379, 1–4 (2015)ADSMathSciNetCrossRefMATH

28.

Zhou, C., Xue, K., Wang, G., Sun, C., Du, G.: Birman–Wenzl–Murakami algebra and topological basis. Commun. Theor. Phys. 57, 179 (2012)ADSMathSciNetCrossRefMATH

29.

Zhou, C., Xue, K., Gou, L., Sun, C., Wang, G., Hu, T.: Birman–Wenzl–Murakami algebra, topological parameter and Berry phase. Quantum Inf. Process. 11, 1765–1773 (2012)ADSMathSciNetCrossRefMATH

30.

Zhao, Q., Zhang, R.Y., Xue, K., Ge, M.L.: Topological basis associated with BWMA, extremes of \(L_1\)-norm in quantum information and applications in physics. arXiv:1211.6178 (2012)

31.

Gottesman, D., Chuang, I.L.: Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature 402, 390 (1999)ADSCrossRef

32.

Nielsen, M.A.: Universal quantum computation using only projective measurement, quantum memory, and preparation of the \(0\) state. Phys. Lett. A 308, 96 (2003)ADSMathSciNetCrossRef

33.

Brauer, R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. 38, 857–872 (1937)MathSciNetCrossRefMATH

34.

Zhang, Y., Zhang, K.: GHZ transform (I): Bell transform and quantum teleportation. arXiv:1401.7009 (2014)

35.

Vollbrecht, K.G.H., Werner, R.F.: Why two qubits are special. J. Math. Phys. 41, 6772–6782 (2000)ADSMathSciNetCrossRefMATH

36.

Kraus, B., Cirac, J.I.: Optimal creation of entanglement using a two-qubit gate. Phys. Rev. A 63, 062309 (2001)ADSCrossRef

37.

Zanardi, P., Zalka, C., Faoro, L.: Entangling power of quantum evolutions. Phys. Rev. A 62, 030301 (2000)ADSMathSciNetCrossRef

38.

Shende, V.V., Bullock, S.S., Markov, I.L.: Recognizing small-circuit structure in two-qubit operators. Phys. Rev. A 70, 012310 (2004)ADSCrossRef

39.

Gottesman, D.: Stabilizer codes and quantum error correction codes. Ph.D. Thesis, CalTech, Pasadena, CA (1997)

40.

Boykin, P.O., Mor, T., Pulver, M., Roychowdhury, V., Vatan, F.: A new universal and fault-tolerant quantum basis. Inf. Process. Lett. 75, 101–107 (2000)MathSciNetCrossRefMATH

41.

Pourkia, A., Batle, J., Raymond Ooi, C.H.: Cyclic groups and quantum logic gates. arXiv:1509.08252 (2015)

Title: Quantum teleportation and Birman–Murakami–Wenzl algebra
Authors: Kun Zhang
Yong Zhang
Publication date: 01-02-2017
Publisher: Springer US
Published in: Quantum Information Processing / Issue 2/2017
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-016-1512-8

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 2/2017

Characterizing error propagation in quantum circuits: the Isotropic Index

Room-temperature spin-photon interface for quantum networks

Toffoli gate and quantum correlations: a geometrical approach

Compact quantum gates for hybrid photon–atom systems assisted by Faraday rotation

Overcoming experimental limitations in a nonlinear two-qubit gate through postselection

Biseparability of noisy pseudopure, W and GHZ states using conditional quantum relative Tsallis entropy