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Entanglement and Density Matrix of a Block of Spins in AKLT Model

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Abstract

We study a 1-dimensional AKLT spin chain, consisting of spins S in the bulk and S/2 at both ends. The unique ground state of this AKLT model is described by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a contiguous block of bulk spins in this ground state. It is shown that the density matrix is a projector onto a subspace of dimension \(\left(S+1\right)^{2}\) . This subspace is described by non-zero eigenvalues and corresponding eigenvectors of the density matrix. We prove that for large block the von Neumann entropy coincides with Renyi entropy and is equal to \(\ln\left(S+1\right)^{2}\) .

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References

  1. Affleck, A., Kennedy, T., Lieb, E.H., Tasaki, H.: Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799–802 (1987)

    Article  ADS  Google Scholar 

  2. Affleck, A., Kennedy, T., Lieb, E.H., Tasaki, H.: Valence bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys. 115, 477–528 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  3. Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517–576 (2008). arXiv:quant-ph/0703044

    Article  ADS  MathSciNet  Google Scholar 

  4. Arecchi, F.T., Courtens, E., Gilmore, R., Thomas, H.: Atomic coherent states in quantum optics. Phys. Rev. A 6, 2211–2237 (1972)

    Article  ADS  Google Scholar 

  5. Arnesen, M.C., Bose, S., Vedral, V.: Natural thermal and magnetic entanglement in the 1D Heisenberg model. Phys. Rev. Lett. 87, 017901 (2001)

    Article  ADS  Google Scholar 

  6. Arovas, D.P., Auerbach, A., Haldane, F.D.M.: Extended Heisenberg models of antiferromagnetism: Analogies to the fractional quantum Hall effect. Phys. Rev. Lett. 60, 531–534 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  7. Audenaert, K., Eisert, J., Plenio, M.B., Werner, R.F.: Entanglement properties of the harmonic chain. Phys. Rev. A 66, 042327 (2002). arXiv:quant-ph/0205025

    Article  ADS  Google Scholar 

  8. Auerbach, A.: Interacting Electrons and Quantum Magnetism. Springer, New York (1998)

    Google Scholar 

  9. Bennett, C.H., DiVincenzo, D.P.: Quantum information and computation. Nature 404, 247–255 (2000)

    Article  ADS  Google Scholar 

  10. Campos Venuti, L., Degli Esposti Boschi, C., Roncaglia, M.: Long-distance entanglement in spin systems. Phys. Rev. Lett. 96, 247206 (2006)

    Article  ADS  Google Scholar 

  11. Chen, Y., Zanardi, P., Wang, Z.D., Zhang, F.C.: Entanglement and quantum phase transition in low dimensional spin systems. New J. Phys. 8, 97 (2004). arXiv:quant-ph/0407228

    Article  Google Scholar 

  12. Zhao, Y., Zanardi, P., Chen, G.: Quantum entanglement and the self-trapping transition in polaronic systems (2004). arXiv:quant-ph/0407080

  13. Hamma, A., Ionicioiu, R., Zanardi, P.: Ground state entanglement and geometric entropy in the Kitaev’s model. Phys. Lett. A 337, 22 (2005). arXiv:quant-ph/0406202

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Giorda, P., Zanardi, P.: Ground-state entanglement in interacting bosonic graphs (2003). arXiv:quant-ph/0311058

  15. Cramer, M., Eisert, J., Plenio, M.B.: Statistical dependence of entanglement-area laws. Phys. Rev. Lett. 98, 220603 (2007). arXiv:quant-ph/0611264

    Article  ADS  Google Scholar 

  16. Cramer, M., Eisert, J., Plenio, M.B., Dreisig, J.: On an entanglement-area-law for general bosonic harmonic lattice systems. Phys. Rev. A 73, 012309 (2006). arXiv:quant-ph/0505092

    Article  ADS  Google Scholar 

  17. Fan, H., Korepin, V.E.: Quantum entanglement of the higher spin matrix product states (2008, in preparation)

  18. Fan, H., Lloyd, S.: Entanglement of eta-pairing state with off-diagonal long-range order. J. Phys. A 38, 5285 (2005). arXiv:quant-ph/0405130

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. Fan, H., Korepin, V., Roychowdhury, V.: Entanglement in a valence-bond-solid state. Phys. Rev. Lett. 93, 227203 (2004). arXiv:quant-ph/0406067

    Article  ADS  Google Scholar 

  20. Fan, H., Korepin, V., Roychowdhury, V.: Valence-bond-solid state entanglement in a 2-D Cayley tree (2005). arXiv:quant-ph/0511150

  21. Fan, H., Korepin, V.E., Roychowdhury, V., Hadley, C., Bose, S.: Boundary effects to the entanglement entropy and two-site entanglement of the spin-1 valence-bond solid. Phys. Rev. B 76, 014428 (2007). arXiv:quant-ph/0605133

    Article  ADS  Google Scholar 

  22. Fannes, M., Nachtergaele, B., Werner, R.R.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144, 443–490 (1992)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Franchini, F., Its, A.R., Jin, B.-Q., Korepin, V.E.: Ellipses of constant entropy in the XY spin chain. J. Phys. A 40, 8467 (2007). arXiv:quant-ph/0609098v5

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Franchini, F., Its, A.R., Korepin, V.E.: Renyi entropy of the XY spin chain. J. Phys. A 41, 025302 (2008). arXiv:0707.2534

    Article  ADS  Google Scholar 

  25. Freitag, W.-D., Muller-Hartmann, E.: Complete analysis of two spin correlations of valence bond solid chains for all integer spins. Z. Phys. B 83, 381 (1991)

    Article  ADS  Google Scholar 

  26. Garcia-Ripoll, J.J., Martin-Delgado, M.A., Cirac, J.I.: Implementation of spin Hamiltonians in optical lattices. Phys. Rev. Lett. 93, 250405 (2004). arXiv:cond-mat/0404566

    Article  ADS  Google Scholar 

  27. Ghosh, S., Rosenbaum, T.F., Aeppli, G., Coppersmith, S.N.: Entangled quantum state of magnetic dipoles. Nature 425, 48–51 (2003)

    Article  ADS  Google Scholar 

  28. Vedral, V.: Entanglement hits the big time. Nature 425, 28–29 (2003)

    Article  ADS  Google Scholar 

  29. Gu, S.-J., Lin, H.-Q., Li, Y.-Q.: Entanglement, quantum phase transition, and scaling in the XXZ chain. Phys. Rev. A 68, 042330 (2003)

    Article  ADS  Google Scholar 

  30. Gu, S.-J., Deng, S.-S., Li, Y.-Q., Lin, H.-Q.: Entanglement and quantum phase transition in the extended hubbard model. Phys. Rev. Lett. 93, 086402 (2004)

    Article  ADS  Google Scholar 

  31. Hadley, C.: Single copy entanglement in a gapped quantum spin chain (2008). arXiv:0801.3681

  32. Hagiwara, M., Katsumata, K., Affleck, I., Halperin, B.I., Renard, J.P.: Observation of S=1/2 degrees of freedom in an S=1 linear-chain Heisenberg antiferromagnet. Phys. Rev. Lett. 65, 3181 (1990)

    Article  ADS  Google Scholar 

  33. Haldane, F.D.M.: Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model. Phys. Lett. 93A, 464–468 (1983)

    ADS  MathSciNet  Google Scholar 

  34. Haldane, F.D.M.: Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis Nel Stat. Phys. Rev. Lett. 50, 1153–1156 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  35. Hamermesh, M.: Group Theory and Its Application to Physical Problems. Dover, New York (1989)

    Google Scholar 

  36. Hirano, T., Hatsugai, Y.: Entanglement entropy of one-dimensional gapped spin chains. J. Phys. Soc. Jpn. 76, 074603 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  37. Holzhey, C., Larsen, F., Wilczek, F.: Geometric and renormalized entropy in conformal field theory. Nucl. Phys. B 424, 443 (1994). arXiv:hep-th/9403108v1

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Iblisdir, S., Latorre, J.I., Orus, R.: Entropy and exact matrix product representation of the Laughlin wave function. Phys. Rev. Lett. 98, 060402 (2007). cond-mat/0609088

    Article  ADS  MathSciNet  Google Scholar 

  39. Its, A.R., Jin, B.-Q., Korepin, V.E.: Entanglement in XY spin chain. J. Phys. A 38, 2975 (2005). arXiv:quant-ph/0409027

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. Jin, B.-Q., Korepin, V.E.: Quantum spin chain, Toeplitz determinants and Fisher-Hartwig conjecture. J. Stat. Phys. 116, 79 (2004). arXiv:quant-ph/0304108

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Katsura, H., Hirano, T., Hatsugai, Y.: Exact analysis of entanglement in gapped quantum spin chains. Phys. Rev. B 76, 012401 (2007). arXiv:cond-mat/0702196

    Article  ADS  Google Scholar 

  42. Katsura, H., Hirano, T., Korepin, V.E.: Entanglement in an SU(n) valence-bond-solid state (2007). arXiv:0711.3882

  43. Keating, J.P., Mezzadri, F.: Random matrix theory and entanglement in quantum spin chains. Commun. Math. Phys. 252, 543–579 (2004). arXiv:quant-ph/0407047

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. Kennedy, T., Lieb, E.H., Tasaki, H.: A two-dimensional isotropic quantum antiferromagnet with unique disordered ground state. J. Stat. Phys. 53, 383–415 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  45. Kirillov, A.N., Korepin, V.E.: Correlation functions in valence bond solid ground state. Sankt Petersburg Math. J. 1, 47 (1990)

    MathSciNet  Google Scholar 

  46. Kirillov, A.N., Korepin, V.E.: Algebra Analysis 1, 47 (1989)

    MathSciNet  Google Scholar 

  47. Kitaev, A., Preskill, J.: Topological entanglement entropy. Phys. Rev. Lett. 96, 110404 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  48. Korepin, V.E.: Universality of entropy scaling in one dimensional gapless models. Phys. Rev. Lett. 92, 096402 (2004). arXiv:cond-mat/0311056

    Article  ADS  Google Scholar 

  49. Latorre, J.I., Orus, R.: Adiabatic quantum computation and quantum phase transitions. Phys. Rev. A 69, 062302 (2004). quant-ph/0308042

    Article  ADS  MathSciNet  Google Scholar 

  50. Latorre, J.I., Orus, R., Rico, E., Vidal, J.: Entanglement entropy in the Lipkin-Meshkov-Glick model. Phys. Rev. A 71, 064101 (2005). cond-mat/0409611

    Article  ADS  MathSciNet  Google Scholar 

  51. Latorre, J.I., Rico, E., Vidal, G.: Ground state entanglement in quantum spin chains. Quant. Inf. Comp. 4, 048 (2004)

    MathSciNet  Google Scholar 

  52. Laughlin, R.B.: Anomalous quantum hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395 (1983)

    Article  ADS  Google Scholar 

  53. Levin, M., Wen, X.G.: Detecting topological order in a ground state wave function. Phys. Rev. Lett. 96, 110405 (2006)

    Article  ADS  Google Scholar 

  54. Lloyd, S.: A potentially realizable quantum computer. Science 261, 1569–1671 (1993)

    Article  ADS  Google Scholar 

  55. Lloyd, S.: Envisioning a quantum supercomputer. Science 263, 695 (1994)

    Article  ADS  Google Scholar 

  56. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge Univ. Press, Cambridge (2000)

    MATH  Google Scholar 

  57. Orus, R.: Entanglement and majorization in (1+1)-dimensional quantum systems. Phys. Rev. A 71, 052327 (2005). quant-ph/0501110

    Article  ADS  MathSciNet  Google Scholar 

  58. Orus, R., Latorre, J.I.: Universality of entanglement and quantum-computation complexity. Phys. Rev. A 69, 052308 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  59. Osborne, T.J., Nielsen, M.A.: Entanglement in a simple quantum phase transition. Phys. Rev. A 66, 032110 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  60. Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to quantum phase transitions. Nature 416, 608 (2002)

    Article  ADS  Google Scholar 

  61. Pachos, J.K., Plenio, M.B.: Three-spin interactions in optical lattices and criticality in cluster Hamiltonians. Phys. Rev. Lett. 93, 056402 (2004)

    Article  ADS  Google Scholar 

  62. Plenio, M.B., Eisert, J., Dreisig, J., Cramer, M.: Geometric entropy in harmonic lattice systems. Phys. Rev. Lett. 94, 060503 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  63. Plenio, M.B., Hartley, J., Eisert, J.: Dynamics and manipulation of entanglement in coupled harmonic systems with many degrees of freedom. New. J. Phys. 6, 36 (2004). arXiv:quant-ph/0402004

    Article  ADS  Google Scholar 

  64. Popkov, V., Salerno, M.: Logarithmic divergence of the block entanglement entropy for the ferromagnetic Heisenberg model (2004). arXiv:quant-ph/0404026

  65. Rico, E., Briegel, H.J.: 2D Multipartite valence bond states in quantum antiferromagnets (2007). arXiv:0710.2349

  66. Ryu, S., Hatsugai, Y.: Entanglement entropy and the Berry phase in the solid state. Phys. Rev. B 73, 245115 (2006)

    Article  ADS  Google Scholar 

  67. Vedral, V.: High temperature macroscopic entanglement. New J. Phys. 6, 102 (2004). arXiv:quant-ph/0405102

    Article  ADS  Google Scholar 

  68. Verstraete, F., Cirac, J.I.: Valence-bond states for quantum computation. Phys. Rev. A 70, 060302(R) (2004)

    Article  ADS  MathSciNet  Google Scholar 

  69. Verstraete, F., Martin-Delgado, M.A., Cirac, J.I.: Diverging entanglement length in gapped quantum spin systems. Phys. Rev. Lett. 92, 087201 (2004)

    Article  ADS  Google Scholar 

  70. Verstraete, F., Popp, M., Cirac, J.I.: Entanglement versus correlations in spin systems. Phys. Rev. Lett. 92, 027901 (2004)

    Article  ADS  Google Scholar 

  71. Verstraete, F., Porras, D., Cirac, J.I.: Phys. Rev. Lett. 93, 227205 (2004)

    Article  ADS  Google Scholar 

  72. Wang, J., Kais, S.: Scaling of entanglement at quantum phase transition for two-dimensional array of quantum dots (2004). arXiv:quant-ph/0405087

  73. Wang, J., Kais, S.: Scaling of entanglement in finite arrays of exchange-coupled quantum dots (2005). arXiv:quant-ph/0405085

  74. Wang, J., Kais, S., Remacle, F., Levine, R.D.: Size effects in the electronic properties of finite arrays of exchange coupled quantum dots: A renormalization group approach (2004). arXiv:quant-ph/0405088

  75. Zanardi, P., Rasetti, M.: Holonomic quantum computation. Phys. Lett. A 264, 94–99 (1999). arXiv:quant-ph/9904011v3

    Article  MATH  ADS  MathSciNet  Google Scholar 

  76. Marzuoli, A., Rasetti, M.: Spin network quantum simulator. Phys. Lett. A 306, 79–87 (2002). arXiv:quant-ph/0209016v1

    Article  MATH  ADS  MathSciNet  Google Scholar 

  77. Rasetti, M.: A consistent Lie algebraic representation of quantum phase and number operators (2002). arXiv:cond-mat/0211081

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Authors and Affiliations

  1. C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY, 11794-3840, USA

    Ying Xu & Vladimir E. Korepin

  2. Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

    Hosho Katsura & Takaaki Hirano

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  1. Ying Xu
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  2. Hosho Katsura
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  3. Takaaki Hirano
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  4. Vladimir E. Korepin
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Correspondence to Ying Xu.

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NSF Grant DMS-0503712.

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Xu, Y., Katsura, H., Hirano, T. et al. Entanglement and Density Matrix of a Block of Spins in AKLT Model. J Stat Phys 133, 347–377 (2008). https://doi.org/10.1007/s10955-008-9617-9

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