Abstract
We present an introduction to the concept of localizable entanglement (LE) with special focus on its numerical computation. LE is an entanglement measure for multipartite systems, which leads naturally to notions like entanglement length and entanglement fluctuations. After briefly reviewing basic properties of LE we present a scheme for the numerical calculation of LE. It is based on the matrix-product state representation of many-body quantum states and the Monte Carlo method. It can be applied both to pure and mixed states. Using this method we calculate the LE of ground and thermal states for various spin systems.
Similar content being viewed by others
References
Bennett CH, Brassard G, Crépeau C, Jozsa R, Peres A, Wootters WK (1993) Phys. Rev. Lett. 70:1895
Preskill J (2000) J. Mod. Optic 47:127
Zanardi P, Wang X (2002) J. Phys. A 35:7947
Osterloh A, Amico L, Falci G, Fazio R (2002) Nature 416:608
Osborne TJ, Nielsen MA (2002) Phys. Rev. A 66:032110
Vidal G, Latorre JI, Rico E, Kitaev A (2003) Phys. Rev. Lett. 90:227902
Calabrese P, Cardy J (2004) J. Stat. Phys. P 06002
Gunlycke D, Kendon VM, Vedral V, Bose S (2001) Phys. Rev. A 64:042302
Ortiz G, Somma R, Barnum H, Knill E, Viola L, quant-ph/0403043
Somma R, Ortiz G, Barnum H, Knill E, Viola L, quant-ph/0403035
Roscilde T, Verrucchi P, Fubini A, Haas S, Tognetti V (2004) Phys. Rev. Lett. 93:167203
Verstraete F, Popp M, Cirac JI (2004) Phys. Rev. Lett. 92:027901
Popp M, Verstraete F, Martin-Delgado MA, Cirac JI (2005) Phys. Rev. A 71:042306
Verstraete F, Martin-Delgado MA, Cirac JI (2004) Phys. Rev. Lett. 92:087201
Jin BQ, Korepin VE (2004) Phys. Rev. A 69:062314
Pachos JK, Plenio MB (2004) Phys. Rev. Lett. 93:056402
Syljuasen OF (2004) Phys. Lett. A 322:25
Dür W, Hartmann L, Hein M, Briegel HJ, quant-ph/0407075
Fan H, Korepin V, Roychowdhury V, quant-ph/0406067
Subrahmanyam V, Lakshminarayan A, quant-ph/0409048
Raussendorf R, Bravyi S, Harrington J, quant-ph/0407255
Hastings MB (2004) Phys. Rev. Lett. 93:140402
Briegel HJ, Dür W, Cirac JI, Zoller P (1998) Phys. Rev. Lett. 81:5932
Dür W, Briegel HJ, Cirac JI, Zoller P (1999) Phys. Rev. A 59:169
Greenberger DM, Horne M, Zeilinger A (1989) in Bell’s Theorem, Quantum Theory, Conceptions of the Universe. Kafatos M (ed) Kluwer, Dordrecht, pp 69–72
Briegel HJ, Raussendorf R (2001) Phys. Rev. Lett. 86:910
Sachdev S (1999) Quantum Phase Transitions. Cambridge University Press, Cambridge
Affleck I, Kennedy T, Lieb EH, Tasaki H (1988) Commun. Math. Phys. 115:477
Fannes M, Nachtergaele B, Werner RF (1992) Commun. Math. Phys. 144:443
Ostlund S, Rommer S (1995) Phys. Rev. Lett. 75:3537
Metropolis N, Rosensbluth AW, Rosensbluth MN, Teller AH, Teller E (1953) J. Chem. Phys. 21:1087
Bennett CH, Bernstein HJ, Popescu S, Schumacher B (1996) Phys. Rev. A 53:2046
Wootters WK (1998) Phys. Rev. Lett. 80:2245
Bennett CH, DiVincenzo DP, Smolin JA, Wooters WK (1996) Phys. Rev. A 54:3824
Bennett CH, Brassard G, Popescu S, Schumacher B, Smolin JA, Wooters WK (1996) Phys. Rev. Lett. 76:722
Vidal G, Werner RF (2002) Phys. Rev. A 65:032314
Horodecki M, Horodecki P, Horodecki R (1996) Phys. Lett. A 223:1
Peres A (1996) Phys. Rev. Lett. 77:1413
Horodecki M, Horodecki P, Horodecki R (1998) Phys. Rev. Lett. 80:5239
Horodecki P, Horodecki M, Horodecki R (1999) Phys. Rev. Lett. 82:1056
Aharonov D (2000) Phys. Rev. A 62:062311
Cirac JI, Zoller P, Kimble HJ, Mabuchi H (1997) Phys. Rev. Lett. 78:3221
van Enk SJ, Cirac JI, Zoller P (1998) Science 279:205
Kraus B, Cirac JI (2004) Phys. Rev. Lett. 92:013602
DiVincenzo DP, Fuchs CA, Mabuchi H, Smolin JA, Thapliyal A, Uhlmann A, quant-ph/9803033
Laustsen T, Verstraete F, van Enk SJ (2003) Quantum Info. Computation 3:64
Dukelsky J, Martin-Delgado MA, Nishino T, Sierra G (1997) Europhys. Lett. 43:457
White SR (1992) Phys. Rev. Lett. 69:2863
White SR (1993) Phys. Rev. B 48:10345
Verstraete F, Porras D, Cirac JI (2004) Phys. Rev. Lett 93:227205
Vidal G (2004) Phys. Rev. Lett. 93:040502
Verstraete F, García-Ripoll JJ, Cirac JI (2004) Phys. Rev. Lett. 93:207204
Zwolak M, Vidal G (2004) Phys. Rev. Lett. 93:207205
Vidal G (2003) Phys. Rev. Lett. 91:147902
Porras D, Verstraete F, Cirac JI, cond-mat/0504717
Verstraete F, Cirac JI, cond-mat/0407066
Pfeuty P (1970) Ann. Phys. 57:79
Murg V, Cirac JI, quant-ph/0309026
Wolff U (1989) Phys. Rev. Lett. 62:361
Takahashi M (1999) Thermodynamics of One-Dimensional Solvable Models. Cambridge University Press, Cambridge
Bethe HA (1931) Z. Phys. 71:205
Yang CN, Yang CP (1966) Phys. Rev. 150:321
Yang CN, Yang CP (1966) Phys. Rev. 150:327
Gu SJ, Lin HQ, Li YQ (2003) Phys. Rev. A 68:042330
Syljuasen OF (2003) Phys. Rev. A 68:060301(R)
García-Ripoll JJ, Cirac JI (2003) New J. Phys. 5:76
Duan LM, Demler E, Lukin MD (2003) Phys. Rev. Lett. 91:090402
García-Ripoll JJ, Martin-Delgado MA, Cirac JI (2004) Phys. Rev. Lett. 93:250405
Haldane FDM (1983) Phys. Lett. 93A:464
Haldane FDM (1983) Phys. Rev. Lett. 50:1153
Campos Venuti L, Roncaglia M (2005) Phys. Rev. Lett. 94:207207
den Nijs M, Rommelse K (1989) Phys. Rev. B 40:4709
Tasaki H (1991) Phys. Rev. Lett. 66:798
Kennedy T, Tasaki H (1992) Phys. Rev. B 45:304
Oshikawa M (1992) Phys J. Condens. Matter 4:7469
Kruis HV, McCulloch IP, Nussinov Z, Zaanen J (2004) Phys. Rev. B 70:075109
Kennedy T (1990) J. Phys.: Condens. Matter 2:5737
Mermin ND, Wagner H (1966) Phys. Rev. Lett. 17:1133
Author information
Authors and Affiliations
Corresponding author
Additional information
PACS
03.67.Mn; 03.65.Ud; 75.10.Pq; 73.43.Nq
Rights and permissions
About this article
Cite this article
Popp, M., Verstraete, F., Martin-Delgado, M. et al. Numerical Computation of Localizable Entanglement in Spin Chains. Appl. Phys. B 82, 225–235 (2006). https://doi.org/10.1007/s00340-005-2069-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00340-005-2069-x